"Let's not" and "Don't let's" can be used interchangeably when you're suggesting something that (in your opinion) you and someone else shouldn't do.
Tell you what, let's not go to the cinema tonight.
Tell you what, don't let's go to the cinema tonight.
Please, let's not have another argument tomorrrow.
Please, don't let's have another argument tomorrow.
In those cases, they're the same but I would say that the first is MUCH more common. I rarely hear the second version and I don't think I use it as a matter of course.
"Don't let us..." as an imperative would be pretty common in various incarnations.
Pick us up at 3pm please. Don't let us wait too long.
Don't let us down. We really need you to do this.
Don't let us leave before the end. We have a nasty habit of doing that!