According to the Cambridge Grammar of the English Language (Huddleston & Pullum 2002), there are two dialect usages within Standard English with regard to let's.
One of these has let as a verb taking a Direct Object us, and then a Catenative Complement, an infinitival clause. The second has let's as a phonologically and syntactically fused item functioning as a Marker of the 1st person inclusive imperative. In this case the plain form of the verb after let's would be regarded as the Head of the clause.
This second dialect allows constructions where the noun phrases following let's function syntactically and semantically as the Subjects of the following clauses:
- Let's you me and Bob go and see the manager together. (grammatical in Dialect B)
Within speakers of this second dialect, there are some speakers for whom the transformation from verb plus Object to Marker is fully complete, and for whom, because let's is no longer a verb, a dummy auxiliary may be used in the negation of the following verb phrase:
- Let's don't be too hasty.
Notice that if be too hasty was a non-finite subordinate clause functioning as the Complement of the verb LET, then this occurrence of don't would be impossible. We don't find such auxiliaries in non-finite clauses.
Huddleston & Pullum say about this last usage, Let's don't, that this is extremely rare and "cannot be regarded as acceptable in Standard English" (p. 935). However, perhaps it should be regarded more as a regional feature of some Standard Englishes. It seems to be relatively common feature for some speakers in Northern America. It is listed in the OED under let's don't:
c. U.S. colloq. let's don't: let's not, don't let's (do something specified or implied).
1854 G. E. Rice Blondel ii. ii. 38 A shabby trick! Let's do n't.
1900 W. F. Drannan Thirty-one Years on Plains & in Mountains xxv. 425 Let's don't talk about that, please don't ask any more questions about it.
1939 D. Parker Here Lies 33 Let's don't think about a lot of Chinese.
1986 New Yorker 24 Mar. 34/2 Let's don't go yet.
2003 Technol. Rev. Mar. 70/2 But let's don't beat up on the pharmaceutical industry.
Ref:"do, v." OED Online. Oxford University Press, December 2015. Web. 12 January 2016.
As you can see this has been around since 1854. You can see fourteen Google-pages of instances from printed books here. If you ignore the first five or so entries, you'll see that these are nearly all from native speakers. Some of them can be considered formal English.
Bill Clinton, whom it seems is a Dialect B speaker, used this frequently in his speeches. You can see two instances here in a speech of his. This is an excerpt from Remarks to the Community in St Louis, Missouri. September 10, 1996:
Let's don't go back. Let's don't go back and adopt an unwise tax program that sounds so great. Oh, I'll give you more money, they say. What they don't say is, I'll give you more money and then we'll have to cut Medicare, Medicaid, education, the environment even more than we tried to cut it before; and the deficit will go up so you'll have higher interest rates. I say let's build that bridge to the future. We don't want to go back to that past, we tried it the first time and we didn't like it.
You can find twelve other such instances in the linked-to book of speeches from Clinton's administration.
The usage of Let's don't won't be regarded as acceptable in Standard English by many speakers. However, it is becoming more frequent. It has been around since at least 1854. It appears in the titles of songs and the speeches of well-to-do and very well educated American presidents. Upper class wealthy lawyers use it in American dramas. It is frequent enough to have been included in the 1992 version of the OED, albeit with the tag "colloquial". It is therefore perhaps best regarded as a dialect feature of some Standard Englishes. But that last bit is just my opinion.