Since some of the rink websites don't list their closing dates, I've contacted them and compiled them here. Dates are subject to change.
CLOSED - South Street Seaport Rink: Closing March 2, 2014
CLOSED - The Rink at Winter Village in Bryant Park: Closing March 2, 2014
CLOSED - Riverbank State Park Ice Skating Rink: March 16, 2014
CLOSED - Ice Skating Rink at Brookfield Place (World Financial Center): March 16, 2014
CLOSED - LeFrak Center at Lakeside in Prospect Park: Closing March 23, 2014
CLOSED - The Rink at Rockefeller Center: Closing April 20, 2014