forget to do sth.意为忘记要做某事
例：I forgot to lock the door.我忘记锁门了.
forget doing sth.意为忘记做过某事
例：I forgot locking the door.我忘记已经锁了门了.
同样,remember to do sth.意为记得去做某事,而remmember doing sth.意为记得做过某事.