Commas before 'nor' and 'because'

I think you have mixed two constructions—neither ... nor and a compound infinitive—by using parts of each. The compound infinitive has dropped out of sight because of a faulty parallelism. Does this seem more clear? "They managed to neither go to the gallery nor take a trip to the shore because John can't stand heat." With the compound infinitive visible now, you can see you do not want a comma before the "nor."

"Because" does not of itself require a preceding comma, but adverbial clauses at the end of sentences sometimes do. In this case I would leave it out.

The sentence is correct as I've rewritten it, but since you are using a casual narrative style, as shown by your contraction, the neither ... nor construction feels a little stilted. I therefore retract the suggested "neither" and would go along with poster 'user405662' and substitute "or" for "nor" while correcting the second half of the infinitive. "They did not manage to go to the gallery or take a trip to the shore because John can't stand heat."