I think we are getting somewhat off track in this discussion! No one is suggesting that we can conclusively prove the Universe is infinite but I'm curious to know why you think 'the burden of proof' rests on the 'case for infinite' side of the argument? It could equally be argued that if one was to suggest the Universe is finite, then evidence for this must be presented.
However, neither of the above arguments I just made are reasonable. Clearly we do not have conclusive evidence either way. What we do know is that we have not reason to doubt that the cosmological principle applies for the extent of our observable universe, for this we do have hard evidence such as the isotropy of the CMB (the anisotropies are of course of great interest, but are very small in agreement with the cosmological principle), and the general isotropy of galaxies and QSO's seen in large redshift surveys.
Given this data then, we can see that if the Universe is finite, it is clearly at least much bigger than the observable Universe, and for all intents and purposes is infinite. Assuming the Universe is infinite makes the equations easier to deal with, since there are less parameters.
I'm not sure how to discussion got to the point of demanding conclusive evidence for the infinite nature of the Universe? Surely this is impossible! We can only ever say that was haven't observed something, we can never have proof
that is doesn't exist. To make a whimsical example, it is not reasonable to demand that someone who says that talking monkeys with tea-pots for hands do not exist provide evidence for their non-existence. All you can do is point to the evidence that we have not observed such things, and it makes evolution a simpler model if it does not have to explain why monkeys should have evolved tea-pots in place of their hands. To further the silly analogy, we could make theoretical predictions about the problems monkeys would face with tea-pot hands, just as we can make predictions about the problems an edge to the Universe would introduce. In both cases our theories may suggest the non-existence of something but to prove
with evidence the non-existence is clearly impossible.
Theories are constructed based upon what is observed, and are made to be as simple as possible given those observations. This is why we assume the Universe is infinite, in full knowledge that this is not absolutely neccessarily the case.