Should I accept this unfunded offer from my dream school or wait for funded offers?
Unless you are independently wealthy, you should never take an unfunded offer for graduate school in mathematics.
Even before the pandemic, some people with a dozen good papers after two successful postdocs - a well above average publication rate - were failing to find jobs.
When I was last broadly applying for academic jobs 10 years ago, the general advice was to send 80-100 applications to jobs you were competitive for in order to have reasonable certainty of getting at least one offer. Even before the pandemic, there weren't that many jobs in total to apply for, and each job gets more applicants than they did 10 years ago. (This year is brutal; the only people getting research-oriented jobs will be AMS Sectional plenary lecture candidates (there are roughly 40 per year, total - that's less than the number of Berkeley+Stanford PhDs per year) or better.)
There are even fewer jobs in industry for which a math PhD is a better qualification than several years of experience.
For a more teaching-oriented academic job, the funded position at a lower-ranked graduate school is actually better preparation, because you will be teaching for your funding, and because the undergraduate population at a lower-ranked school is generally more similar to the undergraduate population at most universities than the undergraduate population at a highly-ranked school.
If you borrow money for graduate school, you could end up in a situation where you are unemployed with no means of paying back your loans. Don't forget education loans cannot be discharged via bankruptcy.
Going to graduate school in math with funding is already a costly decision from the financial point of view, though it's fine if getting to spend a few years doing math is worth it for you (as it was for me). Going to graduate school in math unfunded is potentially ruinous.
Generally speaking, there’s no need to accept an offer as soon as you receive it. Offers have a standard deadline for you to respond. It is normally somewhere on your offer letter. It’s absolutely fine to make further inquiries with HR or one of your prospective advisors regarding funding opportunities:
Can I apply for financial assistance?
Are there any scholarships or teaching assistantships I qualify for?
Can I be funded by your research grant professor X?
All excellent and valid questions that show your genuine interest in enrolling. Graduate school is expensive, and not everyone has the financial privilege of spending several years with no financial support. Admission committees know this and will try to make it easier for good applicants to get some support if possible.
Accepting an offer and then backing out in favor of another is a big no no. Not only did you waste everyone’s time, you also made some other candidate miss an opportunity. People will rightly be upset.
What you can do is ping the schools you’re waiting on. A simple “Hello, I just received a few offers I’m seriously considering and would like to know whether your department had the chance to review my application. I need to make a final decision by XXXXX so would appreciate an answer before that deadline, many thanks!”
You will likely get an answer pretty fast.
Edit: as Alexander Woo mentions in their answer, taking up an unfunded PhD is a bad idea. You may find yourself in a financial hole that you may not be able to get out of for years to come, having to hold off on other important life decisions (buying a house, starting a family) that may look very far away now, but will become frighteningly, depressingly real before you know it. This comment does not hold for Masters degrees: these are often offered on a full tuition deal, but are also a short term commitment.
tl;dr: Agreeing to work for free hurts your colleagues. Don't.
Graduate-level research is important work, which benefits society, the academic institute and the PI/advisor/lab/research group. It merits reasonable compensation - similarly to how you would expect payment seeking other work based on your undergraduate degree.
In many places the term "unfunded PhD" is not used. As a PhD candidate, you are a proper employee of your university or research institute. If you don't finish your PhD within an allotted amount of time, your employment will run out, but doing a PhD while being non-employed is like volunteer work. It's not impossible, but it is unreasonable for you to work full-time as a volunteer, for years.
Unfortunately, graduate researchers, whose work involves both practical contribution and an aspect of traineeship, are in a situation that's easy to exploit - and exploited it is. The struggle for proper employment conditions, and even recognition of status as a contributing employee, is ongoing in many countries.
See also: Why do universities fund Ph.D. students in the sciences?
Now, you might say "Ok, but personally I'm willing to forego a salary and various rights and benefits to get into the research institute I like" (and note: It's not primarily a school, despite the term).
But - it's not just about you. You are hurting your fellow graduate researchers by accepting a position as a PhD candidate researcher without pay. So - you should not do it.
Maybe you don't need the income and can live off of savings, or support from your parents or spouse. But - suppose one of them worked as, say, a nurse or a teacher; and now, the government starts a volunteer program to promote education or public health, and places volunteers as nurses or teachers, in hospitals and schools - for zero pay (This is not a made-up example, this happens in some countries.) I'm sure those volunteers will have the best of intentions; and it is not inconceivable for someone to volunteer to tend to the sick or teach children in their community - but still, in a money-based economy, they would be used as a cudgel to beat down those who rely on these job for a living.
A related question: Can I somehow leverage my willingness to work for a lower salary in job applications?