Even if your APR is 0% for any particular statement billing cycle, there is nonetheless a minimum payment (typically 1% or 2% of the statement balance) that must be paid by the due date marked on the statement (typically 25 days from the statement closing date) to keep the account in good standing. Failure to make this minimum payment in timely fashion can have unpleasant consequences (all laid out in the Terms and Conditions of the 0% APR offer that you accepted). It can be a flat fee or penalty for failure to make the required minimum payment in timely fashion, or cancellation of the 0% APR offer etc.
Moral: Make at least the required minimum payment in timely fashion, or be prepared for the consequences. You don't need to pay the entire statement balance in timely fashion or even make any additional payments of any amount beyond the required minimum payment while the 0% APR offer is applicable, but get at least that minimum paid by the due date, and anything extra can be paid later if you choose to do so.