Probably not, but maybe.
A 2008 study by the National Association for College Admission Counseling (NACAC) reported that 81% of private colleges and 93% of public colleges practiced need-blind admissions. That means your FAFSA® will go to the financial aid office, and the information isn’t shared with or examined by the admissions committee, so therefore won’t at all affect your admissions chances.
That said, there have been informal reports in more recent years that more colleges are “need aware.” That might mean that the admissions team will do the first read-through of your application without looking at your financial circumstances, but later on in the process might look at it. Or it might mean that they’ll look at your finances as “one factor” out of many. Or even that they might only look at your FAFSA® when you’re a “borderline” admissions case (so they’re on the fence about whether to admit you).
In general, though, admissions officers say that fears about financial need affecting admissions are overblown, and that it only affects admissions decisions in a very small minority of cases.
So go ahead and fill out the FAFSA®, without fear!