My dryer, which is probably 16 years old, has suddenly started a squeeking sound when it runs - does it need oil (if so where) or a new belt of some kind?

There are several places the squeek could be coming from. The belt may be slipping (replace the belt). The bearings of the rollers that the drum rides on may be wearing out. You need to pull the drum to replace them. The squeak may also be coming from the motor, but I'd eliminate the other two items, first.