I did two visits this week, and I'm doing another one next week (these are all very close to me, one good thing about Boston). These visits will definitely inform my interview responses.
When I visited the first school this week I was approached by two ad-com people who introduced themselves and we talked for a bit and they were really friendly and I was being charming and sounding intelligent (you know, I used Rhyme's ice maneuver) and had a good chat and made a good impression so I have to imagine that if they happen to be the ones that eventually see my application, it will be a good thing that they remember my conversation with them. (Disclaimer: Maybe I'm totally delusional and they thought I was an idiot, but I thought it went well).
The other school I visited I sat in on a class and the professor happened to be the director of the particular specialization that I'm interested in. He was really friendly and we chatted for a while before class and he said, "Well it was great meeting with you, I review all the applications for the _____ specialization, so I'll keep an eye out for you."
Obviously this all varies by school. I think if you went to HBS after applying they would probably have security escort you out, but other schools are more receptive to student visits, and if you read school websites you can get a sense for the ones that want to admit people that REALLY want to go to their schools (i.e. aren't using it as a backup).