There's a couple of methods to try and I've had successful results with both. The surface that the permanent marker is on is a key factor though. If it's card or paper, then just forget it. It's on there for good. Any shiny surfaces though such as glossy photos and you're in with a (good) shout. Even works on matt photos.
First one is lighter fluid. Don't go crazy, just apply a small amount to a cotton bud, tissue or cotton wool and gently rub the marked area to be removed. If it's not removed in full on the first go just repeat. It will disappear in no time. I find that the best way.
The other thing to try is a dry eraser. Remember those markers and eraser pads they used on the whiteboard at school? One of those. Basically take the (erasable) marker from the set and completely colour over the permanent marker to be removed. Really, just cover it with that pen. Take the eraser, give it a wipe and it will remove everything. I wouldn't recommend using this method on anything but glossy surfaces though, but it DOES work.
I've bought numerous autographs in the past with dedications on them which have been successfully using the above methods.