In short, no, you will not need an EDL after the TSA begins enforcing the REAL ID act.
You will be able to travel with your non-compliant license plus additional identification (see http://www.dhs.gov/real-id-public-faqs: "A driver’s license or identification card from a noncompliant state may only be used in conjunction with a second form of ID for boarding federally regulated commercial aircraft").
You may already have another acceptable identification document (from https://www.tsa.gov/travel/security-screening/identification):
- U.S. passport
- U.S. passport card
- DHS trusted traveler cards (Global Entry, NEXUS, SENTRI, FAST)
- U.S. military ID (active duty or retired military and their dependents, and DoD civilians)
- Permanent resident card
- Border crossing card
- DHS-designated enhanced driver's license
- Airline or airport-issued ID (if issued under a TSA-approved security plan)
- Federally recognized, tribal-issued photo ID
- HSPD-12 PIV card
- Foreign government-issued passport
- Canadian provincial driver's license or Indian and Northern Affairs Canada card
- Transportation worker identification credential
The TSA identification requirement is not absolutely strict. See the above-linked page under "Forgot Your ID?" which states, in part, "... you may still be allowed to fly ... there is no standard list of what alternate forms of ID are acceptable."
The requirement seems very likely to be implemented well after 2016. The actual date has not been announced; it has been postponed repeatedly over the course of several years; and, obviously, the prospect of a large majority of travelers from New York (and New Hampshire, Louisiana, and Minnesota) having to show additional ID and slowing down the ID check is something that the TSA would like to avoid. It's therefore likely that they will keep pushing the implementation date into the future as long as states are making progress towards implementation.
Most importantly, New York State appears to be taking steps toward REAL ID compliance; at least, they have applied for an extension. Certainly, it appears that the state does not intend to force New Yorkers to have to obtain an EDL in order to fly. The DMV website says, for example, that "DMV has requested an extension from the federal government that would allow any New Yorker to continue to use their current state-issued driver license or ID card to board a domestic flight." Remember that it is also in the TSA's interest to reach an agreement that allows New Yorkers to fly with a driver's license, so they are likely to grant the extension.
See also a piece from syracuse.com that quotes a New York State official:
"We have submitted a request for an extension to the REAL ID Act and our discussions with DHS have been very productive," said DMV spokeswoman Casey McNulty.
"We have no reason to believe that any New Yorker will have a problem using their current state-issued ID card to get on a plane come January 2016," McNulty said.
Interestingly, the article goes on to say that it's impossible to learn why exactly New York has been found not to be in compliance.
It also discusses the option of providing a second ID with your non-compliant driver's license, such as "a birth certificate, Social Security card, credit card or marriage license"; however, nobody would confirm whether any particular document would be acceptable.
That's consistent with my experience flying without a listed form of identification. They do admit that it's possible (see above), but they're very vague on the details. I suspect that this is to avoid committing to anything that terrorists could subsequently use to get on board planes for nefarious purposes.
Still, I can confirm that it's possible. My wife, not a US citizen, holds only a foreign driver's license. We flew once from New York to California, and she forgot to bring her passport because, of course, it was a domestic flight. She showed her foreign license, her work ID, and several bank cards. Interestingly, they were rudely dismissive of the foreign license, despite its being by far the most secure of the documents. I don't remember exactly which of the others they accepted, but in any event, they let her fly. (And, despite our worrying about it, she was also allowed to board the flight back home.)
I just found http://www.dhs.gov/secure-drivers-licenses, where DHS says:
DHS has granted an extension to New Hampshire on Friday, October 9 and fully anticipates that it will grant extensions to Louisiana and New York as early as Tuesday, October 13. DHS is working with Federal agencies to incorporate these changes into their REAL ID enforcement procedures as soon as practicable.
Update, October 22, 2015: The DHS page has not yet been modified, but the extension was granted on or before the 14th. At least, NY 1 reported that it was granted early on the morning of the 15th.