Yes, verbs needing different prepositions need each to keep their own: your two example sentences actually mean different things.
The community was not supportive of, or happy with, the result.
The community was not supportive or happy with the result.
The first means that the community did not support the result; the second says it wasn't happy with the result and didn't support something else, because supportive doesn't go with with and therefore it doesn't go with with the result. There could [and probably normally would] be a comma after supportive but it doesn't necessarily need it.
Incidentally you might consider using Neither...nor where there are two negatives: "The community was neither supportive of, nor happy with, the result."
You can get away with not repeating the preposition when it's the same for both verbs. Note the second comma in the examples below though: omitting that will separate built from the care. If you want to make this sort of sentence negative, neither...nor is required.
The Town Hall was built, and the decoration applied, with considerable care.
The Town Hall was neither built, nor the decoration applied, with any care at all.