The 1830 edition changed the singular burden to the plural burdens. This change is probably accidental since there is nothing inappropriate about the singular usage in this passage. The 1837 edition restored the singular by reference to 𝓟. For a similar example where the 1837 edition relied on 𝓟 to restore an earlier reading, see the discussion regarding the inflected form sees nearby in verse 32.
The probable reason the 1830 compositor set burdens is that all eight prior occurrences of the noun burden(s) in the book of Mosiah are in the plural:
There is one additional instance in the text of the plural noun burdens: “that your burdens may be light” (Alma 33:23). Besides five instances of the singular burden in Isaiah quotations, there are two more instances of burden/burthen in the Book of Mormon text:
In Alma 62:29, the 1906 and 1981 LDS editions emended the archaic burthen to burden.
Summary: Maintain the singular burden in Mosiah 29:34 (“that the burden should come upon all the people”).