burn has 17 distinct synonyms spread across 13 of its 20 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.
cause to burn or combust
“The sun burned off the fog”
feel strong emotion, especially anger or passion
“She was burning with anger”
execute by tying to a stake and setting alight
“Witches were burned in Salem”
spend (significant amounts of money)
“He has money to burn”
feel hot or painful
“My eyes are burning”
a browning of the skin resulting from exposure to the rays of the sun
an injury caused by exposure to heat or chemicals or radiation
a place or area that has been burned (especially on a person's body)
Synonyms, antonyms and the broader and narrower relations for burn come from Princeton University’s WordNet. Full sources and licences.