Hi there -- I think I disagree with most of the things that have been said so far. I just appealed a post-secondary rejection at UCSD and now I have an interview. They seemed very nice about it. I thought I was a competitive candidate and wrote them a letter explaining why I thought I would be a good fit. It's more common than you think; some schools (UCSF, for instance) have a whole different committee that reviews applications again because a lot of people do this.
At many schools only one person decides who will get an interview. They are given a packet of 30 secondaries and are asked to pick five or so who will be granted an interview. Sometimes there are two reviews to make it more fair. The reviewer's mood, the group of applications that yours is in, the alignment of the stars, etc., can all play a part in determining whether you get in.
Being persistent is part of being a doctor. If an HMO denies your patient treatment, would you just say, well, okay, you weren't really my first choice of HMO anyway, and can you tell me how I could get the treatment next year? Of course this is a bit of a stretch, but I do think that appealing can't hurt so long as you're gracious about it.
Good luck whatever you do!