Cecil did a fine job of explaining the phenomenon of snake handling, but he left some important information out.
First, the passage to which he refers, Mark 16:17-18, and indeed most or all of Mark 16 (I can’t give the reference for sure; don’t have my Bible with me), is itself a matter of debate among Biblical scholars. The linear notes to the New International Version state that this entire passage does not exist in the earliest and best available manuscripts. Presumably, it was added by a later scribe (and thus it was not a part of Matthew’s original autograph). The passage is included in the King James Version with no annotation of any kind, and to those who believe in “King James Only” it is as much a part of Holy Scripture as John 3:16. (If you want to debate about the whole “King James Only” controversy, please open up your own thread in GD).
So how did this passage makes its way into the KJV? Rastahomie’s theory (some day I’m going to publish a paper about this and make a ton of money) is that a scribe (we’ll call him Scribe A) in early Christian history read the passage in Acts (again, can’t provide the reference) where Paul is bitten by a viper and suffers no ill effects, and then decided to write a commentary on it. The next scribe to copy Scribe A’s work copies Scribe A’s commentary, and then the commentary is copied and re-copied until Erasmus compiles the KJV.
Finally, a word on snake-handling itself. While I find the whole practice repulsive and believe that it puts a black eye on Christianity, a surprising number of people are bitten and live to tell about it. How they pull this off is a matter of considerable debate, but I believe that mind over matter certainly has to play a role.
My .02 (actually, I think this post is worth at least .04!).