General synonyms:
* Forgo
* Abstain
* Refrain
* Do without
* Lack
* Miss
* Deprive
* Deny
More specific synonyms:
* Fast (for food)
* Starve (for food)
* Defer (for something desired)
* Sacrifice (for something valuable)
* Dispense with (for something unnecessary)
The best synonym to use will depend on the specific situation and the nuance you want to convey. For example, "forgo" is a more formal word than "do without," and "sacrifice" suggests a more deliberate choice than "lack."