You might need to take this to a "dry-cleaner" and have them use one of their "solvent" based cleaners...getting oil stains out require more aggressive action.
I think I would try a dishwashing detergent that is known to cut grease (like Dawn--they use it to cut the oil from the feathers of birds and other wildlife caught in oil spills). I would apply it directly to the spots and let it soak for awhile, rinse and wash again in the hottest water that is safe for your clothes. It might take several attempts to get the stains all out.
I agree with Shari, but will add that you may not get them completely out of you dried them in the dryer before you noticed the lipbalm container. My hubby has left his Chapstick or those little Vaseline lip tubes in his pockets a few times over the years so I know if you don't catch them before the dryer, you might never get them completely clean again. But Dawn or other grease lifting dish soap should work wonderfully. I'd advise if you have an HE machine to wash them without added laundry detergent the first time to make sure all soap residue is removed. And hand dry the clothes until you know the stains are minimal.
@Ritafiller I would recommend getting some oxyclean crystals. Soak the clothes in water with oxyclean overnight then wash again, I usually add some more crystals to the wash water. Good luck.
Dawn and hand cleaner from the Dollar General stays in my laundry room. Have saved many items even after they have been dried. Good luck.
As an afterthought...I recently recall reading if you "scribble" on the stain(s) with white chalk, or sprinkle them with baby powder, and leave it sit for a day or two, the oily stains will be absorbed by the chalk or baby powder. I haven't actually tried this yet so I can't vouch for its effectiveness, particularly once the clothes have been washed, but it couldn't hurt to try this too.