John 8:24
24 I said therefore unto you, that ye shall die in your sins: for if ye believe not that I am he, ye shall die in your sins.
This verse teaches that unless a person believes that Jesus is the "I AM" of Exodus 3:14, they will not be saved. Some also argue that this is a reference to the way that God speaks in Isaiah in passages such as Isaiah 41:4, 43:10, 43:13, 46:4, 48:12.
In John 8:24, Jesus says egō eimi ("I am", "I am he", "I am the one"). The Jews, after hearing this, ask Him, "Who are you?" (John 8:25), demonstrating they did not understand this to be a divine claim at all, despite assuming the worst of Him, and the Lord responds, "Even the same that I said unto you from the beginning" (John 8:25).
So, what was Jesus saying to people "from the beginning" in the Gospel of John? That He was the "I Am" of Exodus? Or, that He was Yahweh? In the verse just before the passage in question, He has said "I am from above" (John 8:23). Earlier in the chapter, He said, "I am the light of the world" (John 8:12). And, if one examines the first time that Jesus uses the phrase egō eimi in John's Gospel, they will see Him using it to say that He is the Messiah:
John 4:25-26
25 The woman saith unto him, I know that Messias cometh, which is called Christ: when he is come, he will tell us all things.
26 Jesus saith unto her, I that speak unto thee am he.
In verse 26 above, Jesus says egō eimi (ἐγώ εἰμι), and in context, it is Him identifying Himself as the Messiah. After this first instance, Jesus goes on to use the phrase for the same purpose throughout the Gospel of John (John 8:24, 8:28, 8:58, 13:19, 18:4-6). The Gospel of John also records people specifically looking for a prophet to come (John 1:19-21, 6:14, 7:40), and so Jesus is essentially saying, "I am the one, the one you are looking for, the Messiah".
There is a very similar scene in John 9:35-37, in which Jesus identifies Himself as the Messiah ("Son of Man") in a similar way. The man who had been born blind asks Jesus who the Son of Man is, and Jesus responds, "You have seen him, and the one speaking with you is he" (John 9:37).
Additionally, near the end of John's Gospel, he gives a purpose statement, which says:
John 20:30-31
30 And many other signs truly did Jesus in the presence of his disciples, which are not written in this book:
31 But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.
According to the above, John's thesis is that Jesus is the Christ. That is consistent with Jesus making the "I am" statements throughout, identifying Himself as the Christ. This is what is happening in the passage in question from John chapter 8. Jesus is not, in that passage, claiming to be the "I Am" of Exodus 3:14.