Nothing direct however you can take most bus lines from the bus depot near Union Station-DC to NY-Penn Station (see the bus listings on this page), or Amtrak train to Penn Station. From Penn Station to JFK, take either the NYAS express bus, LIRR to Jamaica or E subway to Sutphin Blvd/JFK and Airtrain to T4. Train from DC is 3.5 hours (regular NE Corridor service) and 45 minutes to one hour+ by LIRR or subway. Bus is about 5 hours (or more depending on traffic) to NY.
Nothing direct however you