They cannot deny the right to vote, no. What they can do is say "We don't have enough to bother counting and it will save us money if we don't". They look at it and say "We have 15,000 votes for McCain, 10,000 votes for Obama and 2,000 votes for various other candidates. We have another 3,000 absentee votes, and even if they all vote Obama, they won't change the outcome, so we won't count them." And they don't. Obviously the numbers are made up right here, but that is how it's justified.
And besides, it's only one third vote that's cast by absentee ballot, so that means the majority is counted, right? Isn't that what matters?
I wish I didn't need this, but I know I will: /sarcasm