I used TurboTax to prepare and e-file my federal and New York State tax returns. I owed money on both and I elected to have the amounts debited from a savings account. When the transactions never appeared I realized that I had given the number of an account that has been closed for a while. Whoops! Should I proactively contact the IRS and the NYS Department of Taxation and Finance, or will they just send me a bill?
They will not send a bill, though there's a chance they will eventually send an accusatory letter. You must proactively pay your taxes.
The simplest route is to send a check to each taxing authority with the respective full amounts due. I wouldn't bother calling them. You could also file amended returns with each containing the correct information. As a general rule, tax advisors tend to counsel against giving bank account information to the IRS for payment purposes (as opposed to refund purposes), both to protect the timing of payment and to make it slightly more difficult for them to seize or lien your account.
If you choose to send a check, you can use Form 1040-V and NY Form IT-201-V. Please triple check your Social Security Number matches your tax return SSN, so they correctly credit you for payment. You may include an explanation of the closed account if you are feeling either fearful or contrite, but if the amount due is paid in full, then neither taxing authority should really care about your error.