In my answer I will be using the New American Standard Bible, 1977 version unless otherwise noted.

The short answer, as expressed by Paul Butler in his commentary on John is:

“John 5:31 and John 8:14 have been ridiculed for years by unthinking critics as “contradictions in the Bible.” The critics, as usual, take Jesus’ words out of context and interpret them, having already decided beforehand what He says. A careful study of the two passages in their respective contexts will show that on both occasions He affirmed exactly the same thing from opposite angles.”

Now, let us consider these two passages in context and in depth.

Jesus stated in John 5:31, “If I alone bear witness of Myself, My testimony is not true.”

While in John 8:14 Jesus said, “ Even if I bear witness of Myself, My witness is true; for I know where I came from, and where I am going; but you do not know where I come from, or where I am going.”

First we need to look at each passage of scripture in context and determine what Jesus said in each passage individually; only then can we compare what he ways in each passage to see if there is any conflict.

In John 5:1 to 17 we read of a miracle Jesus performed in which He healed a man at the pool known as Bethesda in Jerusalem on the Sabbath and, in the process of healing the man told the man to “Arise, take up your pallet, and walk.” John 5:8.

The Jews, seeing the man carrying his pallet scolded the man for carrying his pallet because it was the Sabbath. John 5:10.

Once the Jews determined it was Jesus who had healed the man and that it was Jesus who told him to, “Arise, take up your pallet, and walk” then “the Jews were persecuting Jesus.” John 5:16.

As a note of clarification, the better translation in this phrase of John 5:16 might be “the Jews were pursuing Jesus.”  The word “persecuted” in the NASB means, according to William D. Mounce, “to put in rapid motion; to pursue; to follow, pursue the direction of, (Luke 17:23); to follow eagerly, endeavor earnestly to acquire, (Romans 9:30-31; Romans 12:13); to press forwards, (Philippians 3:12 & 14); to pursue with malignity, persecute, (Matthew 5:10-12 & 44).  We have to remember that, according to Jewish tradition, to “take up you pallet” was a capital offense.  Additionally, as F.F. Bruce points out, “Inciting others to break the law (as they understood it) was worse than breaking it oneself.”  (Bruce, F.F., The Gospel of John.” p.. 126.)  According to Guy N. Woods, “Here, incidentally, is the first indication of open antagonism to Jesus on the part of the Pharisees as noted by John.” (Woods, Guy N., John, New Testament Commentaries. p. 100.)  Further, according to J. Ramsey Michaels, “Here, as at the beginning of the chapter, the imperfect tenses [of the verbs – jrr] are noteworthy. The verb “pursued” … describes a repeated or constant action, a fixed policy of regarding Jesus as a marked man.” (Michaels, J. Ramsey, The Gospel of John, The New Internation Commentary on the New Testament.)  So it appears that this does not indicate physical abuse as the word persecute might imply as much as it does the beginning of a continuous pursuit of Jesus by the Jews, specifically the Pharisees, that eventually culminated in His arrest, “trial,” scourging, and crucifixion.

At that point Jesus told the Jews, “My Father is working until now, and I Myself am working,” John 5:17.

The Jews began, in John 5: 18, “seeking all the more to kill Him, “because not only was He breaking the Sabbath, but also was calling God His own Father, making Himself equal with God.”

Then, in verse 19 to 29, Jesus explains that, while He – Jesus – can do nothing on His own authority (because He, Jesus, has voluntarily submitted His will to that of the Father – Philippians 2:5 to 7) the Father has given the Son all authority: the authority to give life (verse 21), and to judge (verse 22). 

In verse 30 then Jesus again says “I can do nothing on My own initiative. As I hear, I judge; and My judgment is just, because I do not seek My own will, but the will of Him who sent Me.”

He then proceeds to verse 31, “If I alone bear witness of Myself, My testimony is not true.” 

In John 5:31 what Jesus seems to have in mind is the principle of Jewish law that “A single witness shall not rise up against a man on account of any iniquity or any sin which he has committed; on the evidence of two or three witnesses a matter shall be confirmed.” – Deuteronomy 19:15.  Therefore, IF Jesus alone was bearing witness of Himself – from the Jewish perspective scripture in Deuteronomy 19:15 – then His testimony would not be true.

However, here it is as if Jesus is saying to the Jews, “Alright, you want to play games and go by your interpretation of God’s law, I will provide you with more witnesses than just Myself.”

Jesus then provides two witnesses, John the Immerser (verse 33 to 37: see John 1:19 to 34) and the Father (verses 37 & 38: see Matthew 3:13 to 17).

John the Immerser told the Jews in John 1:32 to 34, “I have beheld the Spirit descending as a dove out of heaven, and He remained upon Him.  And I did not recognize Him, but He who sent me to immerse in water said to me, ‘He upon whom you see the Spirit descending and remaining upon Him, this is the one who immerses in the Holy Spirit.’  And I have seen, and have borne witness that this is the Son of God.”  [The word “baptize” properly means to immerse – jrr].  Thus, John the Immerser was a witness to the fact that Jesus was the Son of God and, therefore, His testimony was true.

Then in John 5:36, Jesus indicates He – Jesus – will provide an even greater witness, the Father, “ … the Father who sent Me, He has borne witness of Me.”  It was the Spirit who, in the form of a dove, descended upon Jesus, and it was the Father whose “voice came out of the heavens” saying, ‘Thou art My beloved Son, in Thee I am well-pleased.’” – Matthew 3:13 to 17; Mark 1:9 to 11; Luke 3:21 & 22.

So, here in John 5 Jesus is saying that He IS NOT testifying of Himself, by Himself: He has two other witness, as provided by the Law, John the Immerser and the Father.

Now let’s consider John 8.

First, we have to go back to John 7.  In John 7:1 to 24, we read of Jesus and his brothers – his physical brothers.  Jesus is in Galilee as the Feast of Booths (Sukkot or the Feast of Tabernacles) was approaching and his brothers tell Him to “Depart from here, and go into Judea.”  Jesus sent his brothers to the feast then in verse 10, “He Himself also went up, not publicly, but as it were, in secret.”

The Feast of Booth was a major celebration and commemoration for the Jews which occurred during the later part of September each year.  It celebrated and remembered the temporary dwellings the Israelites dwelt in during their journey through the wilderness from the time they left Egypt until the crossed over the Jordan into the promised land.  For seven days the Jew lived in a temporary dwelling erected for the celebration: these temporary dwelling filled the streets of Jerusalem as both residents of the city and visitors to the city came to remember the journey through the wilderness.  This festival also marked the end of the harvest season and was a time of thanksgiving for all the blessings bestowed on the people by God.

Two events that were part of the Feast of Booths need to be noted here.  First, a ceremony called the "Water Libation Ceremony" or "Simchat Beit HaShoeivah" (Rejoicing at the Water Drawing) is performed.  This ceremony involves drawing water from the Pool of Siloam and pouring it on the altar as a libation along with wine.  This ritual is seen as a prayer for rain, and the ceremony is associated with God's provision of water in the wilderness during the Israelites' journey.  Second is the “Illumination of the Temple” which involved lighting four massive lampstands in the Court of the Women at the temple.  These lights were said to illuminate the entire area, including Jerusalem's streets and the temporary sukkahs (booths) which filled the city streets.

In verse 11 to 13 we see that the Jew were looking for him while the multitudes were divided about him, some saying, “He is a good man” but others were saying, “No, on the contrary, He leads the multitude astray.” but NONE were “speaking openly of Him for fear of the Jews.”

Then in verse 14 we find Jesus goes to the temple and begins to teach.  The Jews then begin to question how it is that Jesus can teach as He does because He has “never been educated” verse 15.

Jesus responds by saying, “My teaching is not Mine, but His who sent Me.  If any man is willing to do His will, he shall know of the teaching, whether it is of God, or whether I speak from Myself.  He who speaks from himself seeks his own glory; but He who is seeking the glory of the One who sent Him, He is true, and there is no unrighteousness in Him.  Did not Moses give you the Law, and yet none of you carries out the Law? Why do you seek to kill Me?”

NOTICE here Jesus is pointing everything back to the Father who sent Him: Jesus is claiming nothing about Himself other than He – Jesus – has been sent by God in verse 16.  Then in verse 17 Jesus explains men do HIS will IF they know the teaching.  The teaching can be either from God or from He – Jesus – Himself because both are teaching the same thing.

Then in verse 18 Jesus explains He is not speaking from His own initiative but from the One who sent Him and therefore, Jesus IS NOT seeking His own glory but rather that of “the One who sent” Him: Jesus then affirms He – Jesus, the one Sent – is “true, and there is no unrighteousness in Him.”

Jesus then poses a question to the multitude in verse 19, “ Did not Moses give you the Law, and yet none of you carries out the Law? Why do you seek to kill Me?”  The multitude then says, “You have a demon! Who seeks to kill You?” – verse 20.

In verse 21 Jesus brings the conversation back to the same idea He expressed in John 5 when He healed the man as the pool of Bethesda by saying, “I did one deed, and you all marvel.”

Jesus then proceeds to explain, “On this account Moses has given you circumcision (not because it is from Moses, but from the fathers), and on the Sabbath you circumcise a man.  If a man receives circumcision on the Sabbath that the Law of Moses may not be broken, are you angry with Me because I made an entire man well on the Sabbath?  Do not judge according to appearance, but judge with righteous judgment.”

In His response in verse 22 to 24 then Jesus is turning the arguments of the Jews and their tradition back on itself: He is showing the inconsistency of man’s ideas verses God’s.  BE CONSISTANT: If you can’t work on the Sabbath by healing a man, then you can’t work on the Sabbath by circumcising one either.

In verse 25 to 27 the people begin to ask, basically, “if this is the man the Pharisees are trying to kill, LOOK! He is speaking in the open publicly so, why are they doing nothing!?”  The people then answer themselves in verse 26, “The rulers do not really know that this is the Christ, do they?”  Verse 30 explains that “no man laid his hand on Him, because His hour had not yet come.”

What follows in verses 32 to 36 is the Pharisees reacting to the multitude and then being confused over the statement by Jesus, “no man laid his hand on Him, because His hour had not yet come.  You shall seek Me, and shall not find Me; and where I am, you cannot come.”

Then on the las day of the feast Jesus cries out, “If any man is thirsty, let him come to Me and drink.  He who believes in Me, as the Scripture said, ‘From his innermost being shall flow rivers of living water.’”  Speaking of the Holy Spirit who would be sent after His ascension back to heaven, verses 37 to 39.  With the Jews just have celebrated the Simchat Beit HaShoeivah, the statement by Jesus here would have been of particular significance.

Then in verses 40 through 52 there is a division within the multitude of people concerning who Jesus is.  The “officers” question those who wanted to seize Jesus, “Why did you not bring Him?”  After a “debate” between the officers and the others Nicodemus asks, “Our Law does not judge a man, unless it first hears from him and knows what he is doing, does it?”  At that point Nicodemus is ridiculed and basically written.  However, Nicodemus was actually referencing Rabbinical Law.  According to Rabbi Eleazar ben Pedath who wrote around 300 AD, “Flesh and blood may pass judgement on a man if it hears his words; if it does not hear them, it cannot establish its judgment.”  Roman law, on this point was the same as seen by the Roman governor Festus explaining, “… it is not the custom of the Romans to hand over any man before the accused meets his accusers face to face, and has an opportunity to make his defense against the charges.” – Acts 25:16.

John 8:1 to 11 describes an incident in which a woman caught in adultery is brought to Jesus who refuses to condemn her saying simply, “Neither do I condemn you; go your way. From now on sin no more.”

Then the narrative turns back to Jesus’ statement in John 7:37 & 38 concerning water, drink and living water but changing the reference to light saying, “I am the light of the world; he who follows Me shall not walk in the darkness, but shall have the light of life.” verse 12.  This, of course was done in the context of the just celebrated light ceremony during the Feast of Booths in which huge lights were lit in the temple that illuminated much of the city and the booths that filled the streets in which the people lived during the festaval.

The Pharisees then said in John 8:13, “You are bearing witness of Yourself; Your witness is not true.” And they are saying basically the same thing they said in John 5:18 when they were seeking to kill Him, that He, Jesus, was making Himself equal with the Father.

The Jews, in verse 13, once again call the principle of Deuteronomy 19:15 into their argument.

Jesus then, despite the explanation He had provided in John 5 where He had provided two witnesses, John the Immerser and the Father, interjects a new argument as Merrill C. Tenney explained here Jesus, “shifted His argument from the basis of abstract legality to the principle of His personal competence.” (Tenney, Merrill C., John, the Gospel of Belief, 1976, p. 144).

In verses 17 & 18 Jesus will, once again, provide both Himself and the Father as witnesses which meets the requirements of the law, both Rabbinical and Roman.  Jesus then give three reasons why the Pharisees argument was unjustified.

• 1.      Because Jesus was and is divine AND because He is going back to the divine realm again, He can bear witness of Himself, verses 18 to 20;
• 2.      Because the Pharisees were judging Him, “according to the flesh” or, as the NIV has it, “You judge by human standards.”  Since Jesus was not merely a man – John 1:1 – the Pharisaical judgement was totally and completely lacking, invalid, and of no consequence.
• 3.      Because His testimony rested on and was supported by the Law itself.  Verses 17 and 18, “In your own Law it is written that the testimony of two witnesses is true.  I am one who testifies for myself; my other witness is the Father, who sent me.”  Two witnesses as prescribed by the Law. (See Lipe David L., John 1-12, Truth for Today Commentarty, pp. 358-59).

So, in both passages of scripture Jesus is saying the same thing: I am my own witness and I have two other witnesses, John the Immerser and the Father, according to the Law which you yourselves are ignoring.

There is no contradiction of conflict between the two verses when read and understood in the context provided for each of the passages. 

In both passages Jesus is showing He and the Father are ONE and therefore in both passage it is not Jesus alone who is witnessing but Jesus and the Father

Just so, you Corinthians, just because you have been converted over to Jesus Christ does not mean you are “fool proof.”  You too can fall from grace (Galatians 5:4 – You have been severed from Christ, you who are seeking to be justified by law; you have fallen from grace.) and when you do you need to repent and be restored, James 5:16 and I John 1:9.