Wiktionary
drink up
vb. to finish one's drink.
WordNet
drink up
v. drink to the last drop; "drink up--there's more wine coming" [syn: drain the cup]
Usage examples of "drink up".
I think he started to drink up his cold coffee, and then wanted a spot of liqueur to take with it.
Puddles were linked by rivulets, and the nearly frozen ground could drink up little of the flood.