For the past 10 months I have lived and worked in DC for a DC-based company. I still own a home in NY. Should I file my 2016 taxes in DC or NY?
You should talk to a tax professional because these rules can get complicated. You'll need to determine if you fit some specific definitions for either state then determine how much, if any, of your income needs to be reported to that state's tax authority. Some of the criteria is very narrow and the way you use your home in NY will impact your tax status in the state (and possibly in the county).
Here's a little reading for you:
If you were a resident in DC (legally defined as living more than 50% of the tax year in DC), you are legally required to file your taxes in DC, regardless of you owning property in NY. At that point, the house in NY is considered an asset, not as a residence.
edit: but agreed with quid, you should talk to a tax accountant.