boire une cigarette

to smoke a cigarette

In Lebanon, boire (to drink) does the job of fumer (to smoke), so someone asking to drink a cigarette just wants a light.

Viens, on sort boire une cigarette avant que la réunion reprenne.

Come on, let's step out for a smoke before the meeting starts again.

Projet Voltaire's M. Akiki suggests it may trace back to the narguile, a water pipe with a water filled base, which made drinking the smoke feel natural, and the verb stuck even for ordinary cigarettes.