Yes. The area of model theory you are asking about is known as "Geometric stability theory". This is a very active area of research. The standard reference is the book by Pillay with the same name.
These connections have actually gathered a fair amount of recent attention (a google search will provide you with links to conference webpages and some articles), one of its highlights being Hrushovski's "The Mordell-Lang conjecture for function fields", J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. But, really, most recent work in model theory is in this area. For example, take a look at the list of publications by Scanlon, for many interesting examples of model theoretic ideas leading to number theoretic results.
Actually, connections between model theoretic ideas and algebraic geometry have been around for a while, starting with work of Abraham Robinson, though it is fair to say that their recent sophistication is due to the deep insights of Hrushovski, Pillay, Zilber, and their students and collaborators. (I fear any list of names I give is bound to be embarrassingly incomplete.)
Similarly, model theoretic work on "$o$-minimality" is connected in serious ways with real-algebraic geometry. There have also been some interesting connections between this area and set theory, mostly due to the fact that real-algebraic geometry gives us some insight on the study of rings of continuous functions and their quotients. You may want to look at "Super-real fields. Totally ordered fields with additional structure", London Mathematical Society Monographs. New Series, 14. Oxford Science Publications; The Clarendon Press, Oxford University Press, New York, 1996, by Woodin and Dales.
In a different direction, Talayco, a student of Andreas Blass, studied connections between cohomology and set theory.
For a completely different approach, though perhaps not exactly in the direction you intend, locales and, in general, topos theory, allow a foundational presentation that people feeling more comfortable with category-theoretic ideas may prefer to the set theoretic approach. The usefulness of the framework lies in part in that it gives us a way to study duality theorems (such as the one relating Stone spaces with Boolean algebras) in a unified fashion. See for example, "Stone spaces" by Johnstone or "Natural dualities for the working algebraist" by Clark and Davey.
N.B. The following was initially an answer to a very similar question, now closed as duplicate. As, imo, my answer suits this question just as well (if not even better), this question is still open and has no answer, and I think that I gave some suggestions that I haven't seen in the possible duplicates, I'm taking the liberty to give it here (with very slight modifications) instead. If this is somehow out of line, let me know and I'll delete it.
First: I am not a set theorist, but I have a BSc in mathematics, and I am almost done with my MSc (also in mathematics), and set theory happens to be one of those subjects outside of what I do, that I find especially fascinating. I have taken a graduate level course in logic with set theory, but I have only looked at the pure course in set theory and forcing. So, my answer is very much from a students perspective, which I hope is a good thing.
When we did set theory as part of a more advanced course in logic (but still only technically requiring a first course in mathematical logic), we mainly used
1.) R. Cori, D. Lascar; Recursion Theory, Gödel's Theorems, Set Theory, Model Theory. Oxford University Press.
This is part II of a series (duology?) of books on logic (first one here). I must say, starting out with axiomatic set theory, I really liked this one, and this is perhaps especially good if you want a somewhat concise yet rigorous introduction. It will also suit you well if you wanted lots of exercises, and it also has solutions to them (which ime is unusual at this level). (The first book is also a very good rigorous introduction to mathematical logic.)
I also really like
2.) Notes on Set Theory, Second edition, Springer 2006, by Y.N. Moschovakis,
which is of course a more complete book on set theory, but includes axiomatics. Both these books are, in my opinion, concise (at least 1), rigorous, yet accessible, but should still be challenging enough.
The main book used in the pure set theory course at our department is
3.) Kenneth Kunen, Set Theory – an Introduction to Independence Proofs, North-Holland 1980
N.B. I have only skimmed this one, however it looks good, and imo, all book recommendations and choices for course literature in logic courses at our department that I've read, has been of very high quality. (We have a very long tradition of logic and many people doing logic related research here, i.a. Per Martin-Löf, so I have great confidence in their suggestions. Also, Here is a review of the book.)
Another book I found useful, that was among the suggested reference literature for the logic course I took was
4.) Thomas Jech, Set Theory, Third edition. Springer 2000
I think both 3 and 4 above are rather standard, but the others may be less well known, and are actually the ones I prefer (however 1 not being purely set theory). You should be able to read the TOC of each of these books through the links I provided above.
It's always hard to know precisely what someone is after when asking these type of questions, but hopefully this will be of some help.
(Just if you're (or anyone is) interested, but perhaps not that relevant: other suggested, perhaps less well known (also seemingly far more challenging), reference literature for the set theory course was
Best Answer
With a strict enough definition of "non-foundational mathematics" I think the answer is probably "no" (although I would be very interested in seeing potential examples.) However, this shouldn't make mathematicians working on such mathematics feel safe about using unrestricted comprehension. The reason is that it's not always clear a priori what mathematics will turn out to be "foundational".
Indeed, people may start working on some mathematics that seems non-foundational but then turns in a foundational direction. For example, Cantor's development of set theory was a natural consequence of his study of sets of uniqueness in harmonic analysis.
If someone working in a supposedly non-foundational branch of mathematics ended up with a contradiction by using unrestricted comprehension, then with the benefit of hindsight we could say that he or she must have been working in an area related to foundations after all.
It might seem like cheating to make such a declaration after the fact, but perhaps it is not: It seems likely that, from a novel use of unrestricted comprehension to obtain a contradiction, one could obtain a novel use of replacement to obtain a theorem that could not have been obtained without replacement (i.e. using only restricted comprehension). I say this because replacement is a natural intermediate step between restricted and unrestricted comprehension.
Mathematics that uses replacement in an essential way is often considered ipso facto to be foundational. So I think it is likely that mathematics that uses unrestricted comprehension an an essential way (to the extent that it can be salvaged) would be considered foundational as well.
(This answer doesn't address the question of how long, on average, it would take people using unrestricted comprehension in non-foundational-seeming areas of mathematics to run into problems. I think that question is very interesting but probably also very hard to answer.)