Do I need to file a 2016 form 8606 if I made a nondeductible contribution in 2016 but withdrew $4000 from my IRA

[Critter]

Yes you do especially if all your contributions were not non deductible. When you entered the 1099-R for the distribution the program asked you if you had any basis in the IRA (prior non deductible contributions) and then you would have entered the IRA contribution later in the program and made the election to make it non deductible.  If you didn't do both then your return is incorrect and should be amended. 

[macuser_22]

If you had a distribution in 2016 you should have received a 1099-R after January 2017 reporting that.  

After you enter the non-deductible in the IRA contributions section *and* when you enter the 1099-R into TurboTax and answer the follow-up questions about the total 2016 year end value of all Traditional, SEP and SIMPLE IRA accounts in existence, then TurboTax should produce a 8606 that shows both the 2016 nondeductible contribution and the taxable/non-taxable portions of the 2016 distribution.   That is also done on lines 6-15 on the 8606 form.