这要分情况讨论。两个词后面既可以加to do也可以加doing.
forget to do something = 忘记去做某事. 例句: I forget to lock the door. 我忘记锁门了.
forget doing something = 忘记做了某事. 例句: I forget locking the door. 我忘记(已经)锁过门了.
remember to do something = 记得去做某事. 例句: Remember to lock the door. 记住要锁门.
remember doing something = 记得做过某事. 例句: Remember locking the door. 记住(已经)锁过门了
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