How would the mathematics relevant to physical theory be different if it didn't use real numbers? Real numbers assume we can have infinite precision and some of the theory behind them uses infinite processes to establish certain proofs. A small band of mathematicians–eg ultrafinitists–disagree with this and deny the possibility of infinite processes or the logical structure of proofs that require them.
I always assumed that this was an interesting but faddish idea.
But I was recently reading a book by the very influential mathematician and computer scientist Donald Knuth (incidentally one of the popularisers and inventors of surreal numbers, a totally different way of establishing a theory of numbers) and though he didn't doubt the theory of real numbers he made the argument that it is wrong to assume they apply to the real world.
He argues (in chapter 6 of Things a Computer Scientists rarely Talks About, bold highlight mine, italics Knuth):

When I say that the question "finite or infinite?" is a red herring, I don't mean simply that philosophers and theologians have often been arguing about an unimportant issues. I also mean that physicists and scientists fail to realise this. For example, take the literature of chaos theory: Hundreds of papers have been written about the behaviour of solutions to unstable recurrences, by people who assume that real numbers are real.


...But it is a tremendous leap of faith to assume that real numbers apply perfectly to the real world...


...It seems to me that a new branch of physics is needed, called maybe "discrete physics" or something like that, to study the effects of the assumption that parameters can be infinitely precise and to consider instead that the universe probably has only a finite–but extremely large–number of states.

I'm not interested in the philosophical implication or speculations, or whether ultrafinitism has a point. But the question of whether we misunderstand some physics because we assume real numbers apply to the mathematics underpinning physical theory sounds about as important as realising that the universe is not based on flat Euclidian geometry.
So what parts of mathematics relevant to physics would be different if the concept of real numbers doesn't work for physical theory? And what are the implications for physics?
 A: I will try an address the question in a couple ways although as you pointed out, this line of questioning inevitably leads down the Philosophy of Physics rabbit hole, which I am not sure members of this SE would appreciate much.
Firstly, what is physics in the most raw sense? It is the generation of mathematical abstractions that can be used to describe patterns that we see in nature. Over the last century there has been tremendous work done in the area of mathematical abstraction; so the empirical basis of physics is often overlooked(String Theory is guilty of this). But in so far as Physical theories go, they must be consistent with our observations and explain future ones sufficiently. So let's look at paradigm shifting discoveries in Physics; did Einstein say Newton was wrong? Well sort of, but not quite, Newton was an approximation. He couldn't have been completely wrong since objects still fall at constant acceleration, and last I checked we're still in an Elliptic Orbit around the sun. Einstein's work changed our view of the world not because of his cool math tricks, but because he developed a framework that could explain two inconsistent physical theories in a unified way, using cool math tricks.
Now let's look at the alternate formulations of Classical Mechanics done by Hamilton, Euler and Lagrange. Their work was at its core math. They did not explain any new phenomena, or expand the laws of physics in a meaningful way; they developed deeper techniques of mathematical abstraction to express Newton's laws with a further level of generality. Is this really useful for Physics? Yes of course! Is it new Physics? ...sadly no (at least I would say so).
I don't want to reduce the role that discoveries in Math plays in Physics, in fact Einstein and the founders of QM made tremendous use of the Math from their predecessors to develop their new theories of reality. But the Math itself was not enough.
Now that we got all of that out of the way. Where does the existence(or lack thereof) of real numbers come into play? Well it doesn't...unless someone gives us a reason to believe it does. Such a question on the impacts of the relationship of the truth of mathematics to our universe is equivalent to asking "does our world obey Math, or do we use Math to describe it", which is a question that dates back to Aristotle and Plato, and we haven't made much progress since. As far as Physics is concerned Math is a tool, and in so far as Calculus (an inherently continuous field of study) and other tools are useful, let's use them to describe our world.
As to their inherent truth, we don't really care (remember Newton didn't even have a rigorous theory of Calculus, he just kinda made it up as he went along). You may have noticed I said "unless someone gives us a reason to" earlier, and what I mean by that is someone has to make use of the Math to make accurate predictions about our world. It may in fact be that real numbers don't exist, Calculus is flawed, and the ultrafinitists have been wrong all along, they can get a Fields Medal for that! But until someone demonstrates a new theory with new predictions about the world using this ultrafinitism, nobody will be getting a Nobel Prize for it.
Note: In Quantum theory the universe actually does have a minimal length, this is because making smaller and smaller measurements requires more and more energy, and at a certain point it will take so much energy that the act of measurement will create a black hole. This finite minimal length shows us that Quantum Theory inherently implies that the Universe is Discrete, but that does not mean we are banned form using irrational numbers that don't 'technically' exist in the real world. This video describes it quite well https://www.youtube.com/watch?v=nyPdIBnWOCM
A: First, of course we could build physics, using only rational numbers. Since all our measurement apparatuses use rational numbers, we could build all theories using them. This possibly would take more unnecessary work regarding precisions and conversions between measurements of different precisions, rounding, etc. It would be not only in engineering but in physical theories, complicating them. In other words, physical theories would be fused with engineering and more complicated.
As to the positions of Knuth regarding underlying physics. It is completely irrelevant to the first part of the answer.
First, it seems he assumes that limited precision means discreetness. This is wrong. As our current theory tells, there is indeed limit on maximum precision and resolution. But this does not imply discreetness. The logarithm of number of possible states of a system is entropy. Our current theories say, entropy is finite. The number of possible "states" is finite but not integer.
