Whether you apply for fellowships has no effect on the admissions decision, because the evaluation process is need-blind. Admissions director Matt Clemons noted on one of the HKS blog posts that because the ad com evaluates on the candidate's "fit" to the program, they don't necessarily factor in the financial capability, and often end up declining students whose tuition might have been fully funded by respective government/external scholarships.
I think they just really want the students to at least apply for funding, as this is the only chance that they could during the two years at HKS, should they be accepted. This being said, I'm off to finishing my app!