The word burden
is related to the word translated demands
(or NKJV: burdensome) in 1 Thessalonians 2:6. Etymologically, both words are related to the idea of weight
; in 1 Thessalonians 2:6, that idea comes to expression in the concept of importance, and here in 1 Thessalonians 2:9 it conveys the idea of a financial burden, as is clear from the context.
9 For you remember, brothers, our labor and toil: we worked night and day, that we might not be a burden to any of you, while we proclaimed to you the gospel of God.