This past January they closed Its a small world from Jan 4-18 to remove the holiday overlay. Haunted Mansion was closed starting Jan. 18th. So, decorations start coming down around Jan 4th and continues for a couple of weeks. It might vary by a day or two. I know most of the decor will still be up on Jan 2. The 31st and 1st are big days. They wouldnt take everything down until after those days are over.
The trees were still up on Jan 2nd this year but I believe they were gone by the 3rd. We were there until the 4th and I know they weren't there the last day of our stay.
Almost nothing was up when we returned on January 4 last year. The California letters were still Christmassy, but during our stay they changed them back. I think after the 2nd they move quickly to take everything down.
In 2004 and 2005, we traveled to DL on New Years day, we then had our first 3 days to fit in all of the Christmas stuff(rides, parade, fireworks), and on our last day we got the regular fireworks show !