When I file my 2016 form 4868 fed App for Auto Extension, I was told that I need not file my form IT 370 with New York State for 2016. Is this true.

No, that is not correct. As you can see below from the instructions for Form IT-370, that a federal extension form is no longer accepted for the New York extension.

General information Purpose – File Form IT-370 on or before the due date of the return to get an automatic extension of time to file Form IT-201. Note: We no longer accept a copy of the federal extension form in place of Form IT-370.

You can file a New York State extension by using the online link here:  NY Apply For an Extension

Was just the 4868 form accepted in 2016?

 I filed taxes by the October 17, 2020 deadline (3 year statute) and they said an extension was never filed and refuse to send my refund.

Was the IT-370 required back in 2016 like it is now in recent years. 

Yes maybe it’s true the IT-370 is required - but it wasn’t in prior years. 

Also if it was required, why did that not show as an option when I file for an extension using 4868 using TurboTax. 

note : my file by mail instructions for both state and  federal have independent mailing addresses for both types of returns and both say deadline : Oct 16, 2017 ..,.. Meaning my 3 year statute should be until 10/16/20 for both state and federal.  

If form 4868 did not apply to both federal and state extensions in 2016, why does my mailing instruction note the deadline as being in October ? And again why didn’t TurboTax offer a state extension link ?

What year did you stop accepting the federal extension form? 

Click this link for instructions on How to File an Extension Form 4868.

States have different requirements for filing a State Extension.  This link gives info on Filing a State Extension.