I was thinking of incorporating some quotes from a famous author into my game. As this isn't a commercial release, is this legal for me to do? It's a short story author, and I want to use some passages from one of his stories. Is this going to a problem, or is there anything in particular I should keep in mind?

Write him, his publisher or his literary agent and ask for permission.

What you're describing might be covered under fair use, if you're only using a very little bit of the story (i.e. a couple of sentences). Any more than that and you're in dangerous territory. Following Oluseyi's suggestion is the only way to be completely safe.

For Uni projects where I think "This soung would be absolutely perfect for this assignment!", I just email the publisher and explain that this is not a commercial project, and will never be used for any financial gain by anyone.
I have not as yet been told that I cannot use it, people are generaly very understanding and cooperative with educational projects, and I assume they would feel the same about a contest like this, given I count this as an educational project anyhow...