cut out has 3 distinct synonyms spread across 3 of its 7 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.
delete or remove
“Cut out the extra text”
strike or cancel by or as if by rubbing or crossing out
“scratch out my name on that list”
cease operating
“The pump suddenly cut out”
1 of the 7 recorded senses carries no synonym, antonym or related term, so it is not shown above. Every sense, with its definition and examples, is on the full entry for cut out.
Synonyms, antonyms and the broader and narrower relations for cut out come from Princeton University’s WordNet. Full sources and licences.