Never owned a Mac computer...but if you have to worry about leaving it on for a week, then you might need a different kind of computer.
I leave my computer on 24-7...365 days a year...and it's about 4 years old. Some webcams will tend to "freeze" after prolonged use, but I think it's probably more of a driver issue than hardware. But since Mac is supposedly superior, I'm sure they don't have those driver issues.
So that's the long way of saying keep your computer/cam on, as it won't add extra wear to your computer. Just like at all other times, make sure air can move around/through the computer so nothing overheats. You might get one of those cans of compressed air and make sure the fan and everything on the computer is clear of dust bunnies, etc. Enjoy your trip with the added security of iCam! Isn't it awesome?!