Do you need chains anywhere in Washington?

Do a quick search for "Snoqualmie Pass". 

Washington State or the District of Columbia?

Yes, chains are required at times and places during the winter. Drivers are required to carry chains with them in case conditions worsen and they're required to install chains during a "chains required on all vehicles" notice. See: https://www.wsdot.com/winter/tires-chains.htm .Stevens Pass and Snoqualmie Pass are the most likely places where they'd be needed.

Anywhere east of the Cascades. Very much like northern Nevada or eastern Oregon. Cold in the winter mostly sagebrush some patches of trees.

You're worried about driving in snow in Washington state or negotiating the poo in Washington DC?

As for Washington state, it will depend on where and when you are going.  The mountain passes are the only places where you might encounter snow.  Any chain requirements will be for trucks, not for passenger vehicles.  

If you are driving I5, you won't encounter snow.  If you drive I90 over Snoqualmie pass you might encounter snow.  Just slow down, give ample room between you and the vehicle ahead of you and you should be fine.  If your scared stop and wait for the DOT to clear the roads.