1. forget to do sth 忘记要去做某事。 (未做)

2. forget doing sth 忘记做过某事。  (已做)

例句：

The light in the office is stil on. He forgot to turn it off.
 办公室的灯还在亮着，它忘记关了。(没有做关灯的动作)
He forgot turning the light off. 
 他忘记他已经关了灯了。    ( 已做过关灯的动作)
Don't forget to come tomorrow.
 别忘了明天来。         (to come动作未做)

典型例题
 ---- The light in the office is still on.
 ---- Oh，I forgot___.          
  A. turning it off  B. turn it off  C. to turn it off  D. having turned it off
  答案：C。由the light is still on 可知灯亮着，即关灯的动作没有发生，因此用forget to do sth.
而forget doing sth表示灯已经关上了，而自己忘记了这一事实。此处不符合题意。