Elite, wealthy schools often attract students whose parents don't need financial assistance, but almost all still have student-aid budgets to spend, which means that their admissions offices aren't likely to discriminate against those who need grants to pay tuition. Thus, these schools can truly afford to be what's called "need-blind" in their admissions, and checking the financial aid box on the application likely won't hurt your prospects --- after all, many of these schools are looking to attract at least some degree of socioeconomic diversity.
Many universities claim to be need-blind, but in practice this isn't always the case. One criterion that factors into how colleges are ranked is the admissions ratio --- or the proportion of students who are admitted to a university who choose to actually enroll there. So for colleges worried about moving up in ranking, admitting more students than they can support with limited financial aid budgets would be counterproductive, as this would diminish those proportions. If a student declines a school's admissions offer because the school doesn't offer any financial aid, in other words, it makes the school look bad in collegiate rankings, so these students are sometimes rejected instead.
Very few universities admit to factoring financial assistance into admissions, though some do. The Chronicle of Higher Education, for instance, in 2010 quoted the vice president of admissions, financial aid and enrollment at Union College in New York as admitting that, at least at the margins, his school does take aid into account. It measures the need of its applicants against its available funds to ensure each of its needy enrollees can get at least some assistance. So at Union, checking "yes" on the application's financial aid form may, in fact, hurt your chances at admissions --- at least compared to a similarly qualified student who doesn't need the school's help.
It's hard, if not impossible, to know how your answer to the financial aid question will affect your child's application. That's because it's not always clear to what degree schools consider financial aid requests. It doesn't hurt to ask: Call the admissions office and see if it will tell you how it factors in financial aid. Regardless of the answer, or whether you believe it, the best advice is simply to tell the truth. If you need aid, say yes. If not, say no. Saying no shouldn't preclude you from access to merit-based scholarships or government-backed loans.