heave has 15 distinct synonyms spread across 10 of its 14 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
utter a sound, as with obvious effort
“She heaved a deep sigh when she saw the list of things to do”
move or cause to move in a specified way, direction, or position
“The vessel hove into sight”
bend out of shape, as under pressure or from heat
“The highway buckled during the heat wave”
Synonyms, antonyms and the broader and narrower relations for heave come from Princeton University’s WordNet. Full sources and licences.