From Longman Dictionary of Contemporary Englisheat up phrasal verb1 EATto eat all of something Come on, eat up, there’s a good girl.eat something ↔ up She’s made a cake and wants us to help eat it up.2 eat something ↔ up informalUSE/CONSUME to use a lot of something, especially until there is none left Big cars just eat up money.3 be eaten up with/by jealousy/anger/curiosity etcVERY to be very jealous, angry etc, so that you cannot think about anything else → eat→ See Verb table
Examples from the Corpus
eat up• We were always taught to eat our vegetables up.• Margaret ate it all up and then asked for more.• Non-energy saving light bulbs just eat up electricity.• The V8 is a very powerful engine, but it really eats up gas.• My rent eats up most of my money.• Come on, boys -- eat up your supper and get to bed. 