When it comes to drinking, there are plenty of ways to say "drink up" without using those two words. Here are some synonyms for "drink up":
1. Guzzle - to drink quickly and eagerly.
2. Quaff - to drink deeply and heartily.
3. Sip - to drink in small quantities or at intervals.
4. Imbibe - to consume a liquid, especially alcohol.
5. Slurp - to drink noisily.
6. Swig - to take a large gulp of a drink.
7. Chug - to drink a large amount of liquid quickly.
8. Sup - to drink something warmly, slowly and pleasantly.
9. Gulp down - to swallow quickly.
10. Drain - to drink or consume the entire content of a glass or bottle.
With these synonyms, your request for someone to "drink up" will sound more eloquent and varied.