For the most part it wont hurt anything to fly with it on. A few planes, most notably the P38, react badly in certain situations if you have it on. Otherwise, the average pilot probably benefits more from having it on than having it off.
Where it hurts you the most, IMO, is that if you rely on the CT all the time, you forget or simply dont bother to setup controls to manually trim your plane. There are times when its a huge advantage to being able to quickly adjust trim manually.
I dont have all the fancy programmable buttons that alot of the other guys do, so I've learned to make do with using the CT like alot of the others do, as a "bump" to get everything aligned. In a fight, I like to have in on usually, but I DO have manual controls handy for the elevator trim. I used to fly 109s alot, and still do from time to time. If you do, you'll notice that it compresses more than most planes in the game. Having the ability to trim the nose up when your controls are otherwise locked is very handy, and can keep you from making yourself an embarrassed pancake. It can also come in handy in those instances where you find yourself stuck down on the deck and involved in one of those twisty-turny fights for your life.