The Hebrew in the rest of this passage is difficult to translate, but the general meaning is clear if the interplay of the two meanings of burden
is maintained.
The first implication of this interplay is that anyone who claims that their message is from the Lord by simply saying The burden (oracle) of the Lord
will face the Lord’s judgment. This warning applies directly to the false prophets who were claiming to speak on behalf of the Lord but proclaimed a message of peace despite Judah’s unfaithfulness. But this warning is also directed at the whole nation because they have rejected the Lord’s true burden
(oracle) of judgment, as spoken by Jeremiah. They have mocked Jeremiah as being unnecessarily negative and even treasonous in his message of a Babylonian invasion and exile. His message of judgment is an unnecessary burden
(heavy load) for them. In an attitude of self-righteousness (they) call the Lord’s character into question for speaking judgement on his people
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34 And as for the prophet, priest, or one of the people who says, ‘The burden of the LORD,’ I will punish that man and his household.