NY State 414H is 'tax exempt', and CANNOT be e-filed (according to NYS law). But TurboTax states that an e-prepared NYS return must be e-filed. Can I mail my NYS return?

Yes. There is no penalty for mailing a NYS return, and even though NY State says that you have to efile under certain circumstances, they provide a mailing address here: https://www.tax.ny.gov/pit/file/return_assembly_mail.htm

I solved the same issue with a HACK.

Here was my case - I work in NY,  My wife works in NJ and had a 414H in her 'NJ'  W2  ( she works at the University) 

Turbotax throws a NY state efile error based on the 414H being categorized as 'Other' in NJ W2 ( So Stupid !!) 

I found after a million edits  to MY New York - I figured the issue was the NJ W2..   So I went to my wife's w2 and changed  414H to 414H-NJ    and VIOLA  !!!   I was able to e-file NY State !!!!   

Worked like a charm, thanks a bunch!

Thanks so much. It worked for me