Unfortunately, it appears that The Winds of Winter won't be published before Season 6 of Game of Thrones airs on HBO. There was a lot of hope and speculation that it would be published early in 2016 so that fans had a chance to read it before the show aired. Sadly, that isn't going to happen.
On January 2, 2016, George R.R. Martin posted on Live Journal announcing that the book was not done and that it wouldn't be published before the show aired.  According to his post, his publisher told him they would need the book by October of 2015 to get it published in time. When he failed to meet that deadline, they pushed the date back to December. Unfortunately, he missed that deadline as well.
He candidly discussed his failure to meet the deadline, saying:
"Unfortunately, the writing did not go as fast or as well as I would have liked. You can blame my travels or my blog posts or the distractions of other projects and the Cocteau and whatever, but maybe all that had an impact... you can blame my age, and maybe that had an impact too...but if truth be told, sometimes the writing goes well and sometimes it doesn't, and that was true for me even when I was in my 20s."
He goes on to discuss how the publisher even moved the date back to the first of the year, telling him that if he could get the book done by then they could speed up production to get it published before the show aired. Again, however, he missed that deadline. He has no excuses, instead saying:
"Here it is, the first of January. The book is not done, not delivered. No words can change that. I tried, I promise you. I failed. I blew the Halloween deadline, and I've now blown the end of the year deadline. And that almost certainly means that no, THE WINDS OF WINTER will not be published before the sixth season of GAME OF THRONES premieres in April (mid April, we are now told, not early April, but those two weeks will not save me)."
Moving forward, it is hard to say when the book actually will be published. The author didn't make any excuses for missing the original deadlines but also stated that he won't set another deadline for completion, simply for the fact that they cause him too much stress. He said:
"But I won't make excuses. There are no excuses. No one else is to blame. Not my editors and publishers, not HBO, not David & Dan. It's on me. I tried, and I am still trying. I worked on the book a couple of days ago, revising a Theon chapter and adding some new material, and I will writing on it again tomorrow. But no, I can't tell you when it will be done, or when it will be published. Best guess, based on our previous conversations, is that Bantam (and presumably my British publisher as well) can have the hardcover out within three months of delivery, if their schedules permit. But when delivery will be, I can't say. I am not going to set another deadline for myself to trip over. The deadlines just stress me out."
So, the short answer is that no one knows when the book will be published. The only thing that we know for sure is that it won't be published before the show airs in April. Fans are understandably disappointed  but the fact remains that George R.R. Martin is only human.