Here's today's Bushism of the Day, "the president's accidental wit and wisdom," from Slate:
"So we'll bring our ideas, they'll bring theirs, let's clarify the differences, let's don't say bad things about our opponents."
Whoops, sorry, wrong President -- that's actually from President Clinton. The Bushism of the Day today is really this:
"Let's don't just talk about it. Let's actually do it, by passing the legislation."
"I tell people, let's don't fear the future, let's shape it." -- Omaha, Neb., June 7, 2006
As best I can tell, the only supposed flub -- the only supposed humor -- here is "let's don't." (Without that, the phrase isn't terribly rich in content, but neither are "the only thing we have to fear is fear itself," or a wide range of other perfectly normal exhortations from political leaders.)
Yet it's a flub only in the sense that departure from the standard Northeastern/West Coast elite spoken English is a flub. If you search for "let's don't," you'll find it used routinely in spoken English, chiefly (as best I can tell from my searches) by people from flyover country.
The only usage guide I could find that discusses "let's don't" is "The Columbia Guide to Standard American English", which reports, "There are three negative idioms: Let's not stay, Don't let's stay, and Let's don't stay. [I infer that 'stay' is just an example here, and the idiom equally works with other verbs.] All are Standard, although Let's don't is more typically American than Don't let's, which is more typically British." Sounds right to me.