It isn't necessarily about the institution itself, but the faculty. Sure, you will have a much higher density of famous researchers at a place like Harvard or Rockefeller. But even some low tier schools have a handful of well-known researchers. Yes, of course it matters if you do research with renowned faculty who will write you letters for residency - and it's even better if they are from a famous institution. But realistically, most DO students won't be able to get into the research game like that, so you should just focus on getting published anywhere.
Some real world advice - famous researchers at prestigious institutions aren't necessarily the best mentors. You should definitely try and work with someone who is well-known in their respective field, but it is much more important that the researcher is a great mentor and has a track record of mentoring many students like you. There are famous (and not-so-famous) researchers who will do nothing for you as a medical student, even though they may be exceptional at training and mentoring their residents/fellows and colleagues. On the flip side, there are famous (and not-so-famous) researchers who will be happy to take the time to teach you knowing that you are a medical student, and will take great pleasure in going through research topics with you in detail until you understand things at a high level. These are great people to work with. They understand what your goals are and will be glad to help you out, and you will probably get a good letter out of it for residency. The tough part is finding good people to work with. The worst thing is to begin research at a prestigious institution and not have any direction and zero idea what the hell is going on.