The gospels of Matthew and John differ over whether John the Baptist is or is not Elijah. Here's Matthew 11:11-14:
Matthew 17:11-13 is similar. However, John 1:21 is as follows:
Now then. This is not a direct contradiction, as it's possible John the Baptist either doesn't know he is Elijah, or is simply lying about it. However, if neither of these options is appealing, there is little left but to say that either Matthew or John (the author) got it wrong.
However, infallibilists generally respond to this with all sorts of complex doctrines about how John is Elijah, but only in a certain sense. For example, Tektonics writes that John 1:21 really means "I am not Elijah in the sense that you think of it". This is a huge cop-out. The verse says nothing of the sort, and the general problem with infallibilism is that it's forced to make this sort of contrived explanation far too often.
It would be remiss of me not to mention Luke 1:17:
This gives a possible escape: rather than being Elijah, John "goes in the spirit of Elijah" (whatever that means). Note though that Matthew contains no such caveats: for him, John just is Elijah. As for John (the author), he surely expected his readers to understand that the Baptist was not Elijah, in any sense. I therefore feel that the problem is Serious.