I always wonder when letters of interest became so popular among applicants. When I was applying, letters of interest did not exist. You showed interest by paying for the secondary application/fee and going to interviews.
If you think about it, letters of interest are not legally binding, and people send multiple letters of interests to multiple schools. So how is it helpful? Now if it is an update letter (eg FYI, I just got back my grades from last semester, all As, and my article was accepted for publication as a first author in Science), then I can see how it can influence things. But letters of interest saying "oh, I am still interested" doesn't really help (esp since it's not binding and people are starting to send them to multiple places, which dilutes the purpose/intent of letters of interest)
And I'm starting to see it show up in residency/fellowship applications. So far it has no bearing (and not even mentioned) during our rank list meetings (for this year's fellowship ranking, and in prior years for residency ranking since this year's ranking hasn't occurred yet)