Here Oliver Cowdery initially wrote in 𝓟 “I will also ease their burdens”. He was probably influenced by the third person them in the immediately preceding clause: “and I will covenant with this my people and deliver them out of bondage” (at the end of verse 13). Right before that third person usage, the Lord addresses the people of Alma in the second person: “Lift up your heads and be of good comfort / for I know of the covenant which ye have made unto me”. And here in verse 14, the Lord returns to the second person usage (“and I will also ease the burdens which is put upon your shoulders”). Ultimately, the their is quite impossible here in verse 14. Oliver caught his error virtually immediately and crossed out the ir of their, giving the correct the (the ink level for the crossout is unchanged). The critical text will follow the corrected reading in 𝓟 (and the reading in all the printed editions).
Summary: Maintain the definite article the before burdens in Mosiah 24:14; the initial their is a simple scribal error prompted, it would appear, by the preceding occurrence of them in verse 13.