Does applying for financial aid hurt your college admissions chances?

It depends on if the school is need-blind or need-aware.

Most schools in the US are need-blind for US Citizens and Permanent Residents, this means that if you are a US Citizen or Permanent Resident then irrespective of whether or not you apply for financial aid, your application will be treated the same way i.e. your ability to pay for your education is not considered when looking at your application. So if you get in, you will be able to attend. They will ensure you can attend as long as you get in. I can confirm that they will do everything they can to ensure you can attend. I have multiple friends here at Brown who are on a full-ride and for one of them, Brown even gives them money for personal expenses. So this is what need-blind means.

Unfortunately, however, most schools in the US are need-aware for International Students (who aren't US Citizens) and so your ability to pay for your education will be considered when looking at your application. I think this essentially means that if they have to pick between 2 equally deserving applicants but one can pay and the other needs aid, then they will pick the applicant who can pay. This is disheartening and makes it much harder to get into many of the top schools if you are applying for aid, but it certainly is not impossible and can be done and I know many who have done this and even managed to secure full rides from many colleges, including one friend who managed to do so with Brown (even though Brown is known to be stingy with aid).

Additionally, Harvard, Yale, Princeton, MIT and Amherst are the only 5 schools in the US that are need-blind for International Citizens as well. Hope this helps, and get back to me with any queries that you have, I know financial-aid in particular is always confusing and difficult are to navigate.

It depends upon the college/university that you are applying to. Applying for aid at need-blind colleges/universities (colleges which do not consider your family's income as a factor of admission) does not affect your chances of admission, but it does affect your chances if you apply to need-aware colleges/universities.

The following is the list of colleges/universities which are need-blind for financial aid for international students:
1) Massachusetts Institute of Technology
2) California Institute of Technology (only for Regular Decision)
3) Princeton University
4) Harvard University
5) Amherst College
6) Yale University

The following is a list of colleges which are need-aware but are generous for financial aid:
1) Dartmouth College
2) Union College
3) Williams College