Synonyms for Eat Up (same or very similar meaning)
WordNet sense 1 (enclose or envelop completely, as if by swallowing):
bury, immerse, swallow, swallow upWordNet sense 2 (finish eating all the food on one's plate or on the table):
finish, polish offWordNet sense 3 (use up (resources or materials)):
consume, deplete, eat, exhaust, wipe out, run through, use upFrom the ODE community, based on WordNetadd/edit