I have been recently accepted into graduate school for my PhD in Mathematics. I feel very prepared and excited. However, my route is somewhat untraditional as I am "going back." That is, I am walking away from a well paying job. (I should mention that I am unmarried with no children.) I still have a lot of debt from undergrad. And, I am about to accumulate more (although an assistantship will help defray some costs). Obtaining my PhD and teaching at a college is my dream. As such, I am taking the plunge-no matter the cost. Working with the assumption that I will be about $100,000 in debt when I graduate, is it feasible to "live" even if I secure a tenure-track position? For instance, can one "request" to teach intersession and summer sessions for extra income (as a professor)?
Since nobody else has offered a direct answer to the title question...
Should I take on debt in order to get a Ph.D. in mathematics?
If a PhD program isn't willing to waive your tuition and offer you a stipend (in the form of an assistantship or fellowship), they don't really want you. Go somewhere else. If they decide not to give you more funding after a semester or two, you're done. Go somewhere else.
In most math PhD programs in the US, your tuition costs are covered by either the department or your advisor, and you receive a stipend on top of that to cover your living expenses. While it may not be enough to cover the lifestyle you are accustomed to as a working professional, it is certainly enough that you should not need to accumulate substantial debt as a doctoral student! Moreover, you should be able to put any subsidized loans you have in deferral during this time, so that they do not accumulate interest during this period.
With respect to income as an assistant professor, what exactly is permitted depends on the regulations of the particular institution at which you are working. At most schools, however, you are entitled to have "consulting contracts" which would supplement your salary as a faculty member; however, such contracts and the workloads associated with them normally have to be declared to the university. In addition, at many schools, you are paid only a nine-month salary; you can "earn" the remaining three months through external grants and other means (perhaps teaching summer courses, and so on).
Having a lot of undergraduate student loan debt and expressing the amount in dollars makes it sound like you are in the U.S. On the other hand, I haven't heard of a U.S. math grad school admitting anyone in September, so I'm puzzled by where you are. I'll answer assuming it's the U.S., but this answer may not apply in other locations.
As aeismail pointed out, U.S. Ph.D. programs in mathematics fund their students at a level that should let you make it through grad school without new loans. If such a program admits you but won't provide funding, then that shows a serious lack of confidence in you. It amounts to saying that you're welcome to try if you can afford to pay for everything, but they don't think it's a good use of their money.
It seems from your question and comments that you've been admitted now but will not learn about funding until spring. I would strongly recommend applying more broadly this fall, so you can see what your other options are. If the current school does not come through with funding in the spring, then you should view it as just barely being admitted (and you should factor that into your estimate of the likelihood of getting a good job - they might be totally wrong in their judgment of you, but this would indicate that they don't expect you to be one of their more successful students).
I'd recommend taking your favorite among funded offers of admission and treating unfunded offers as tantamount to rejection, but of course it depends on how risk averse you are.
Working with the assumption that I will be about $100,000 in debt when I graduate, is it feasible to "live" even if I secure a tenure-track position?
Try a student loan calculator. The details will depend on your interest rate, repayment schedule, etc., but for an order of magnitude estimate, you'd need to pay roughly $1,000/month to be debt-free in ten years. Whether this is feasible depends on how frugal you are and how much you might make. I've linked to American Mathematical Society salary surveys. You should be careful not to get too optimistic: fancy jobs can pay starting salaries in the $80,000 to $90,000 range, but few people get such jobs and in any case typically after a postdoc. A typical starting salary for a tenure-track job at a teaching-oriented college is $55,000, and getting a tenure-track job at all is certainly not guaranteed.
For instance, can one "request" to teach intersession and summer sessions for extra income (as a professor)?
Sometimes. If you find a job at a research university with a low teaching load, then deliberately increasing the teaching load would be a bad way to get tenure. If you end up working somewhere with a high teaching load, then you might not want to increase it, but I'd expect that you could take on summer teaching if you need the money. It may not pay very well, though.
Just my 2 cents here, but in my opinion, going into a math PhD which is not financially covered 100% more or less; meaning tuition and basic living expenses are paid for, sounds like a bad move to me. Even full teaching assistantships are paid very badly.
One can spend a lot of time in a PhD program. You said that after you graduate, you want to be a math teacher. Well, math teachers don't get paid that much.
If you aren't fully covered under your current arrangements, I suggest you try and find an institution that will fully cover you.
Having said that, you don't give a lot of details about what your financial arrangements during grad school will be, or details about your current debt. If you gave more details, people might be able to offer more concrete responses.
I think that asking about such things in a personal finance forum such as Personal Finance & Money would also be a reasonable thing to do, as this is partly a finance question.
If your dream is to do a PhD in math then go for it by all means, but take into account that you are not putting yourself in the best situation. Could you delay your entry into the PhD program to pay half of the debt with your current good-paying job? You could delay and apply to other programs as well to see if you get more funding. I mean by all means pursue your dreams, but please take into account the financials of it. Also, it might not be your dream after all, once you have spent a few years in the program. The reality might be different than the dream, and for that reason I would suggest working a couple more years, paying off the loan and then entering with a better situation the PhD program.
In any case, you should be able to pay the loan after the PhD straight after graduating, just as if it was part of your mortgage, but you'll have to make some sacrifices, and I assume there is the possibility of you getting married, having more responsibilities but also possibly another source of income. Please take this into account when making your final decision.