Green’s Dictionary of Slang

1. (US Und.) to bring pressure upon someone.

1937		(con. 1905–25) E.H. Sutherland Professional Thief (1956) 9: Other mobs use rip-and-tear (crude) methods that heat up (cause danger in) the spot.
1937		E. Anderson Thieves Like Us (1999) 38: That back there might heat us up a little.

2. (US Und.) of the victim of a confidence trick, to make a fuss or call in the authorities.

1940

D. Maurer Big Con 272: The mark [...] does not ‘heatup’ the train or boat immediately.

3. (US Und.) of pressure of discovery or suspicions, to intensify.

1940		D. Maurer Big Con 273: They cannot change ships as soon as things heat up.
1955		B. Schulberg On the Waterfront (1964) 257: Things were heating up all over the harbour.

4. (orig. US Und.) to infuriate, to annoy.

1937

(con. 1905–25) E.H. Sutherland Professional Thief (1956) 175: Cannons do not generally beat coppers, for it would heat the coppers up against cannons.

5. to arouse sexually.

1975		T. Berger Sneaky People (1980) 161: The little piss-willie who delivered groceries [...] whom she was idly amused to heat up.
2007		R. Milward Apples (2023) 30: The main thing was getting my end away rather than heating her up.

6. (US) of a restaurant, to become busy.

2000

G. Pelecanos Shame the Devil 72: You know I get too nervous when this place heats up.