OK here's an argument I get into all the time. Is it important? to me, yes. I like immersion, and if a game has no basis in reality, it's hard to get immersed in it.
The inevitable answer I get to this is: "But you're playing in a game that has INSERT UNREALISTIC THING HERE! How can you expect realism?"
The answer is simple: if it's necessary for the premise to the game, it gets an automatic pass. Making a zombie game? you need zombies. That's fine. But if you have a powerup that gives me the ability to fly and fart lightning bolts, there had better be some plot explanation of how magic suddenly started happening, otherwise, I'm gonna call zombie bulls**t on you.
Also, realism can be sacrificed in order to increase fun. I'm fine with that, too. Just pick the appropriate sacrifice. An Example: The first time I played Bioshock, I was plagued by the widescreen bug. (At the low graphics settings I had to use, the game didn't handle widescreen monitors well, and you couldn't aim worth a damn.) I didn't know this bug existed, and so I thought my inability to aim was by design, to make the game more realistic. This would be a VERY poor decision on the part of the game designers, particularly since they had also decided to give you a rejuvenation tank that everyone but you was too stupid to use. Being, as Yahtzee calls it, a hybrid of man and refrigerator makes a game more fun. it's unrealistic but a lot more fun than dying every 12 seconds. Having the game be impossible, but Death doesn't matter also is unrealistic, and is NOT fun, and that would be a very poor sacrifice to make.
So yeah, realism is important, but it's less important than fun. It is an element that can be used to make a game more fun, particularly for people who like immersion games. But like any game element it can be overdone to the point that it hampers fun. And that's bad.