Probably the most common antonym of "signing a contract" is "in breach of contract," although "to breach" could also be used. Another common term would be "to violate the contract," as "violate" is often used when rules or laws are broken. If you're not doing something wrong, but merely trying to change what's in the contract, you may "renegotiate" or "modify" the contract. And if you want to get fancy, you could say you "reneged on" the contract (renege is pronounced "reh-NEG," with the second syllable rhyming with "egg"), although this usually has a more negative context than simply "in breach of" or "violate" and is typically used more informally.

Breaching a contract would be the antonym of following the terms of the contract. The opposite of signing a contract (entering into it) is simply not signing it. Or possibly terminating/ending it.