Well, I think whoever you're playing against can see your IP address if they looked for it (I don't think the game is relayed through battle.net. I think it just operates as a matchmaking service). And I guess theoretically, if Blizzard wasn't careful, there might be ways to overflow buffers and inject code. So it's probably not cleared for use on computers in the CIA or anything. But realistically, no, there aren't any security issues to worry about. You aren't downloading executable code. You're just playing a game, which is talking with another game.
Actually I don't know of any security issues with any games ever. So the above is purely theoretical. You just generally don't get hacked through a game. The worst that could happen is you piss off some l33t hacker somewhere and they get your IP and do things to you using that. Like the original Windows XP, out of box, could have visible messages sent to it if you knew the IP. But they fixed that really quick (now you have to enable it explicitly).
If you have any sort of modern OS, most of the stupid things hackers can do with an IP won't work, anyway. There's still a DoS attack, I guess. Not sure if it would work on a consumer PC behind a router. But it's comparatively expensive since you need lots of computers and lots of bandwidth to do that. They can't log in to your computer or anything, unless you've specifically enabled your router and OS to allow remote connection. Which you probably haven't. And even then, just pick some half decent passwords and you're fine.
People do get their battle.net accounts hacked from time to time, but that's because they fall for some sort of phishing and give out their password and account. If you're careful to not give the password to spoof sites you're fine.
So short answer: no.