I am in the process of fixing my own dryer. The symptoms: initially, a squeaking that would last for a couple minutes then go away after the dryer warmed up. Then, over time, it would squeak loudly the entire time the dryer ran.
This is caused by the "rear bearing" going bad, or getting worn down.
I have completely disassembled my dryer to repair this, and I'm at the midway point. Mine is a stackable w/d, but the principle is the same for all dryers.
I'll post complete details on the repair as soon as I complete it and know it's done right.
Until then, I'll tell you this: if your dryer is squeaking, look for the parts manual for your model. Then, look for the place where the rear bearing is mounted. There is a little ball bearing, about 1/4 inch, that rotates against the spindle that mounts your dryer drum. If your dryer is squeaking and you can find this little bearing, just squeeze some grease on it and it will probably run quietly.
The ball bearing is held in by a small plate, about 1 inch X 2 inches, on the back of your dryer. It is centered on the axis of rotation of the drum, so you should be able to find it. Get some heavy duty grease (or get the lube they sell at appliance parts places) and squeeze a bunch under and around that little plate, especially in the center of it where the little ball bearing is, and I'm betting most of the squeak will stop immediately. If it's really bad, though, you'll have to replace the bearing and the cradle for the bearing. The cradle is a teflon cup that supports the drum. As it wears out, it allows the drum to rotate a bit off center, causing further wear.
The only way to fix that is to pull out the drum, replace the cradle, grease the heck out of everything, and hope for the best.
More on this in my next post. If I can, I'll post photos.