The Washington Monument will be there a very long time. It's all load-bearing stone masonry, 15 feet thick at the base. It's not far above sea level, so I guess its foundations could be threatened by rising seas, but stone foundations should do just fine submerged in briny water.
Currently major repairs are done every 30 years or so "to address stone masonry deterioration mechanisms". The 2011 earthquake and hurricane did some damage. "[T]he aspect ratio and manner in which the Washington Monument was originally constructed may have rendered it uniquely vulnerable to ground motion associated with an earthquake." Quotes from https://www.nps.gov/wamo/upload/Post-Earthquake-assessment12... .
I can't project how 200 years of no maintenance will affect the building, especially as there's a difference between a building which is safe for visitors and one which is whole but unsafe.
