tl;dr It should be fine, if not necessarily comfortable. Depending on your budget and interest in roughing it, an airport hotel room would be a reasonable alternative.
From Sleeping in Airports and specifically from their Washingon Reagan Airport sleep guide:
While Washington Dulles [sic] Airport is technically open 24-hours, the Security checkpoints and Secure areas do close for the evening, likely by 11:00 PM. Even if you’re already inside Security, you may have to relocate to the pre-Secure area.
Since you asked in comments, that basically means you'll have to go out through the security checkpoints: you won't be able to stay "in the lounge where the plane lands".
Most seating here is equipped with armrests, so don’t expect too much comfort. Maybe travellers report sleeping on the floor, particularly near the heat vents next to the windows. A sleep mat or sleeping bag will provide some much-needed cushion.
Airport security don’t seem to mind overnight campers, as long as you stick to the pre-Security areas at night.
The site has more details about temperature (generally cold), brightness (generally bright), and other tips.
