No. This is anecdotal, of course, but I went to a public city college for undergrad and ended up at a fully-funded PhD program at an ivy league. My transcript, CV/work experience, letters of rec, GRE scores, personal statement, and very strong background in several modern languages/Greek & Latin were more important than the reputation of my undergrad (though it does have a good reputation for classics, however mediocre the rest of the university is). Don't be discouraged. Faculty members have NEVER given me a problem or treated me differently because of it - it is usually other graduate students who make snide remarks.