For example, the special theory of relativity doesn't work if space-time has any "lowest scale" (i.e. if the distance between any two events can be expressed as n * min_dist, where n is an integer and min_dist is a constant that has the same value for all observers), it you can't infinitely sub-divide it, since then lengths couldn't contract the way that the Lorrentz transforms predicts. Lots of math in QM and QFT also requires space and time to be continuous for other reasons as well.
Also, none of our current laws put any kind of limit on the life span of the Universe as a whole, so space-time is also infinitely increasing, even if heat death or big crunch models are correct.
Of course, none of this is provable per se - it is fundamentally impossible to run an experiment in finite time that proves whether something is finite or infinite. It could be that SR and QFT are wrong in their continuous space assumptions, but only in ways that are too subtle to measure. Still, as it stands, we don't have any theory of how the world works which allows for a discrete space-time, even as an approximation - we need space-time to be continuous for our current theories to work at all.
I always find it interesting nobody seriously questions why something so extraordinarily basic as geometry pervades "the universe". I'd say it's much more likely we are mathy creatures and cannot do anything but perceive logical building blocks somewhat reminding me of a guy using a flashlight proclaiming the universe comprises various configurations of spherical lights.
I can imagine infinity just fine and I'm sure many useful things can be inferred from it, but that doesn't mean it exists. Lots of math is possible but does not exist in reality and that's fine, I'm just being honest about the very real boundaries here. I get the feeling many people confuse the map for the territory.
That's because your imagination is bounded by being a monkey that has spent its entire life in a finite environment — no offense. The universe, however, isn't bounded the limits of our imagination or intuition.