What I've found that works best for me to remove burn marks off a plastic bathroom sink is to sprinkle some baking soda onto the burn marks.
Then dampen a wash cloth with some vinegar and scrub the burn mark or you can also use a toothbrush or other bristle cleaning brush.
The vinegar and baking soda will usually remove the burn mark from the plastic bathroom sink if it's not too severe.
If the burn mark is in the bowl of the sink then you could add some baking soda and undiluted vinegar to the sink and let it work on the burn mark.
Then scrub the burn mark again and it should come off.
Some Dawn Dish Soap will also sometimes remove the burn marks if the burn mark is not very bad.