Not really — unless they do, seems to be the frustrating but honest answer.
“They are one more part of the folder. They are not a significant factor in the vast majority of cases,” says one of Johns Hopkins’ admissions officials — but then he turns around and notes that there have been situations where borderline students had an amazing interview that pushed them over the edge… as well as rare cases where a “arrogant or nasty or ill-informed” student’s interview tips the scale in favor of a “no” vote.
Where it really seems to make a difference is at smaller schools, where admissions offices closely track a student’s communication with (and visits to) a school. A prospective student who visits campus, sits in on a class, and opts for an optional interview seems more engaged and interested than someone who just e-files an application. (Does this result in a socioeconomic bias in favor of students whose families have the means to make college visits? Some admissions officers think so.)
Really, though, it all depends on the school. Amherst stopped offering interviews a couple decades ago; Bard has an “Immediate Decision” program where candidates participate in a faculty-led seminar, interview with an admissions officer, and find out whether or not they got in the next day. Ultimately, it seems like the best thing to do is to shine those shoes, practice your handshake, and be as charming as you can — whether it’ll get you anywhere is another question.