The answer is yes and no.
As an International student or temporary worker, you can sign-up and drive with Uber as long as you have your social security number (SSN).
Uber pays you via direct deposit (once a week) and reports your income to IRS. At the end of the Tax year, you'll receive a 1099-MISC form.
The Yes part:
If you made less than $600 in a tax year, you are not required to file taxes to IRS. You will still get a 1099 tax form but you won't have to fill it.
The No part:
If you are going to make more than $600, you'll have to file taxes and it will be on record.
Now, as others have noted, being on F1 or H1B visa, you are not suppose to work off-campus or for another employer who doesn't hold your work visa.
So by the letter of the law, you are NOT suppose to work. (Even for up-to that $600.)
The real question is: can you drive and not be in trouble?
The answer is: In many cases YES. But there is always that "what-if?"
My (non-legal) advise on this: "don't drive". But if you must for whatsoever financial reason, do it and there is high probability that you will be just fine.
For example, many IT workers have been doing the Amway business, selling on Amazon & eBay, and making money and getting 1099-MISC/1099-K. They have been filing taxes with the IRS and have got their Green Cards (GC) and Citizenship in due time.