From the DC Department of Public Works:
Gift wrapping paper, holiday catalogues, gift boxes, and greeting cards are all recyclable.
(Washington, DC) The holiday season* offers many opportunities to recycle, above and beyond the usual paper, plastic, glass, and metal, according to the District Department of Public Works (DPW), Office of Recycling. Rather than trashing those piles of gift boxes, tags, wrapping paper, and wrapping paper rolls, put those items directly into your recycling cart or bin. Don’t forget that all those gift catalogues, holiday cards and shopping lists are recyclable too when you’re finished with them. For more recycling information, call (202) 645-8245.