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I burned out while pursing a PhD in mathematics but have a masters, and am currently pursuing a masters in philosophy with the hope that it may show continued interest in academia before I re-apply to competitive PhD programs in either mathematics or philosophy (of math or logic etc.) I'm currently teaching calculus 3 and a basic math course as an adjunct, and it looks like this will be my schedule next semester as well. Where I am currently teaching (a private four year university) it looks like I may be teaching a basic philosophy course next semester as well.

[Side note: I've taken my time away from the math phd to get a high GRE and math subject GRE score; what I'm most afraid of is my "burn out" in the math program and distancing myself from that. This is, in part, why I pursued the philosophy program.]

I certainly don't think it can hurt, but what do graduate committees think of teaching experience? Is it advisable to get a letter of recommendation from the chair of the department you teach at when applying to a PhD program (in addition to my other letters from people I was a student of)?

This question is targeted to anyone, but especially those in mathematics, philosophy, and of course philosophy of science and philosophy mathematics folks.

It depends significantly on the particular PhD program you are applying to.

Where I am, teaching experience would be considered a significant positive. It's not our main reason for having a graduate program, but one reason we have a graduate program (and perhaps the only reason we are allowed by the university to have a graduate program) is to have cheap labor for teaching some of our lower division courses. We certainly appreciate having better teachers rather than worse ones, and experience helps. Also, such experience indicates that you are not delusional about the outcomes from our PhD program and are aware that most academics jobs (if you are so lucky to get one) require teaching, frequently lots of teaching, of at least acceptable quality.

On the other hand, at a top ranked program with high-powered research, it probably won't help so much, and some professors may even consider it a slight negative because it indicates you might be interested in devoting time to something that is not research (and there is probably corporate work that is more research-like than teaching as an adjunct.)

Actually, your question is a bit moot since you already have that experience. The more important question is how you present yourself and your existing background to any admissions committee or professor.

Of course, you should also consider what sort of academic career you want to have. Do you see yourself doing state of the art research primarily, or would you be more comfortable as a teacher with research a smaller part of your overall program. For a research directed career and the sort of graduate program you need to achieve it, the teaching would be a relatively small consideration, so I wouldn't stress it as much as if you want a teaching career, in which case it becomes an important element of your overall goals.

The fact that you burned out may be an issue or not. Many people suffer a bit of it along the way to success so it isn't foreign to them. The fact that you stayed connected to the profession after leaving is a plus.

Since you seem to be on the cusp of math and philosophy the ideal program for you may be relatively small, in which case the personal element becomes more important. Even in a large place, however, the number of faculty with similar interests may be a small subset of the whole department(s) so you can still work at a smaller/lower level, rather than an institutional, level to gain acceptance. Say by contacting the department or a single professor, rather than just applying through the normal channels.

But the most important thing, is how you present yourself, both your background and your goals. Everything can contribute to it.