IT has nothing to do with TurboTax.

NJ will do the direct debit if they process your return in time to do it on that date.

Now you must wait several days to see what happens.

Has your NJ return been accepted?  If so, it can still take the state several days to pull your tax payment from your bank account.

Yes.  It's been accepted.  It is now one week since the date I had requested the money to be removed. So maybe it's just a matter of them processing it and even though they accepted it they didn't process it?

You will have to contact the NJ Department of Treasury for more information on this. 

NJ Department of Treasury

NJ is slow. One year, it took them two weeks to pull my tax payment from the date I specified.