How do I remove mildew from paneling without damaging the finish?

These days it is quite clear that mildew is much more than a nuisance. Have a professional look at it. If it is active the best suggestion may be to remove the paneling, find the source and kill it (whatever you need to do) and then replace the paneling with new.

A commercial mildew remover is the best bet.

Ask an employee at Home Depot and they'll match you with the right product.

An old home remedy for removing mildew from fabric is to use lemon juice. Apply lemon juice, and allow to dry. (If it's dry-clean only then apply lemon juice, sprinkle with salt and allow to dry.) Always do a test spot first.

Good luck.