Whats the difference?
How would you draw timelines to clarify the difference?
How would you contextualise these structures?
Cheers
Hi jmckenzie1983
Welcome to the forum. 
Regarding your question about “remembered”, I would describe the difference (specifically in the past tense) this way:
He remembered to do it means that he did not forget to do something he was supposed to do.
He remembered doing it has the sense that he was able to “see” a past activity in his mind. Possibly he might be reflecting on the same activity as mentioned above (i.e., something he didn’t forget to do). Or possibly he was just reminiscing about, enjoying, or reliving a specific memory. Or maybe he had a temporary memory loss and when his memory returned, he suddenly realized he’d done something that he hadn’t been able to remember doing before. :lol:
I doubt there would be any kind of “time line” you could create to illustrate the difference. You might interpret the sense of “to do” (an act) as being of shorter duration than “doing” (an activity), but this could end up being misleading in the context.
Amy