It has been reported that the songwriter seems unsure what it means, himself. He says that it is a song that "just happened". It was inspired by some surrealist films and literature, so it is no surprise that many of the lines are open to multiple interpretations. So probably either of your interpretations and everything that has been said in the other answers is perfectly possible. If we take it simply as an English grammar question though, the written form would be important, because the difference between your two interpretations would probably depend on punctuation. Your first interpretation would be more likely if it was punctuated like this: Take a load off, Fanny, take a load for free. Without the comma after "off", then it is more likely to be asking someone else to lift Fanny's load. But, as a song, the original form is probably oral, and we can't be sure about the punctation.

The load is the burden of obligation. Fanny asked him to do something, and in doing it he ended up doing a lot more than he intended.

Its talking about pressure or stress more emotional than an actual load. Its a metaphor. He is telling her to relax. I think both Jimmy's and Nathan's answers to be valid in the last verse he says Fanny sent him there to give her regards. I used to listen to this song in 1970's on the radio and never understood it beyond face value He went to Nazareth and met some characters. I always understood the chorus to mean relax, like "take a load off your feet and sit down" Its a surreal song but we went with the mood of it not the logic.