The link provides a fine state-by-state list, however I note that it doesn't quite cover the requirements. Here is the text from the GA extension. It is important that recognize that these are going to be the same for any state that accepts the federal extension as a state extension.
"A FEDERAL EXTENSION WILL BE ACCEPTED AS A GEORGIA EXTENSION IF: (1) THE RETURN IS RECEIVED WITHIN THE TIME AS EXTENDED BY THE INTERNAL REVENUE SERVICE, AND (2) A COPY OF THE FEDERAL EXTENSION(S) IS ATTACHED TO THE RETURN WHEN FILED. NOTE: THERE IS NO EXTENSION FOR PAYMENT OF TAX, INCOME TAX OR CORPORATE NET WORTH TAX MUST BE PAID BY THE PRESCRIBED DUE DATE TO AVOID THE ASSESSMENT OF LATE PAYMENT PENALTIES AND INTEREST."
#1 There is no extension for payment (you can always pay more than you think you will owe and get a refund when you file).
#2 To be considered "timely" don't miss the due date on the extension. Then you will be penalized for more late months.
#3 Include the federal extension form as sent to the IRS. Yes, the states can tell if you created one after the face and send it.
The penalty and late fees for non-payment of owed taxes is significantly more than a late filing penalty. That is, if it is a time issue and not a money issue, you are better taking a guess to the high side (so that you know you don't know money) and then filing your taxes.
My experience in Georgia was that the penalty for the late return was negligible and I was refunded the difference promptly. That being said, don't stretch out into years late. Past the point of a normal extension, the agencies (IRS/state DOR) calculate your taxes for you based on whatever information they have and that becomes a difficult number to get them to change. Their information typically will include W-2, 1099s, mortgage interest paid, your filing status assuming no change, but not much else on the deduction side, for example real estate taxes paid, or business expenses.
