What does it mean when you ask me to enter the date I will pay the balance due on my NY State income tax return? What is "Filing Date"?

TurboTax is detecting that you owe New York state taxes, and is asking when you would like to pay the tax due.  The deadline to make this payment is the same as the deadline to file: April 18, 2017.  You can choose to file and pay your New York state taxes prior to this date.

Your filing date is the day you file your tax returns.