Particularly for highly selective universities? Duke, Dartmouth etc.
So you are not a US citizen. It will not affect admissions but it will affect financial aid.
Are we talking about a US citizen who doesn’t have a passport (I suspect pretty common)? Or a non-US citizen without an American passport?
Actually for many top universities, it will affect admissions. Many have a hard or soft cap on the percentage of international students accepted and, as a result, acceptance rate for intl students is substantially less than for domestic students.
@saillakeerie a non-US citizen without an American passport
@skieurope Jeez - Would ED make the process slightly easier for non-US citizens?
Schools allot a set number of seats to international applicants. It’s finite and thus, the stiffer competition. Especially for those schools known for generous financial aid to international students
It’ll be slightly harder if you don’t need financial aid. Significantly harder if you do.
It also depends on which country you’re applying from. If you’re applying from an underrepresented country you might actually be at an advantage. If you’re applying from a place like India or China you’re not doing yourself any favors!
But there are hundreds of very good schools where being a full-pay international student is very welcome. Only at the most selective schools will you be at a disadvantage.
Thank you all for your responses 
I didn’t see anywhere on FAFSA/CSS where they differentiate “US citizens” and “permanent residents (= no American passport)”. So if you are a permanent resident, not an American citizen, I see no differences in terms of FA. I didn’t fill out a common app or any other applications, so I can’t speak in regard to admission, but I am guessing they don’t differentiate between ‘American citizen’ and ‘permanent resident’ either.
Here are the non-US citizens eligible to submit the FAFSA
https://studentaid.ed.gov/sa/eligibility/non-us-citizens
Also, I don’t think the OP said he/she had Permanent Resident status. If so, then he/she would be absolutely eligible for FAFSA and would NOT be evaluated as an International applicant.
OP doesn’t live in the US.