The DAR Museum has a fine quilt collection. Unfortunately, the next exhibition featuring quilts will not begin until April. Their historic room displays may contain a quilt or two, not, I fear, what you're hoping for.
Do explore their web site under past exhibitions for a taste of their holdings. For anyone headed to DC in the future, check the current schedule as they do feature quilts quite often.

don't know what they've got now, but National Museum of Women in the Arts http://www.nmwa.org/ has had fiber art & quilt exhibits.
Florence

FabricWhisperer has given a great summary. The only place I would add is a drive north to Baltimore to the Maryland Historical Society,

http://www.mdhs.org/museum/collections.html

with their collection of Baltimore Album Quilts.

The weather here is gorgeous right now!

Ok..unless you have a car.. for shops there is nothing I am aware of in DC proper. I can give you a list of shops outside DC

G Street Fabrics in Rockville http://www.gstreetfabrics.com/
The Quilt Patch in Fairfax VA http://www.quiltpatchva.com/
The Jinney Beyer Studio in Great Falls VA http://www.jinnybeyer.com/
The Artful Quilter in Burke VA http://www.artfulquilter.com/
Capital Quilts in Gaithersburg MD http://www.capitalquilts.com/

Also of note:
Cottonseed Glory in Annapolis MD about 45 minute drive from DC
http://www.cottonseedglory.com/
Tomorrow's Treasures in Bowie MD http://www.tomorrowstreasures.info/
These two can be done in one day...they are close.
Annapolis is nice for a day trip...you are by the Bay and it is the home of the US Naval Academy. You can take a tour and also do historic Annapolis.

Material Girls in La Plata MD http://materialgirlsquilt.com/site/
It is further out from Annapolis (south) but there is a lovely tea room around the corner. The shop is wonderful. It is really a day trip.

And Sew It Goes in Savage Mill, MD http://www.andsewitgoes.net/
It is located in a historic mill...http://savagemill.com/ which has some lovely shops and a great french bakery.

I am sure I am forgetting other shops.

For exhibits: There is the Smithsonian but not sure what they have out.

Also the Textile Museum http://www.textilemuseum.org/ again I am not sure what they have out

In Baltimore: The Baltimore Museum of Art always has a collection of Baltimore Applique Quilts http://www.artbma.org/exhibitions/index.html

While in Baltimore...I would also visit the American Vision Art Museum...http://www.avam.org/ I know the founder and owner...it is an AWESOME place. It is a gem in Baltimore. No quilts...but cool stuff.

Enjoy your trip!!

FW

Hi gang -
My wife and I will be traveling to Washington D.C. in a couple weeks. Are there any "must see" quilt exhibits or shops that you'd recommend? I'd really appreciate your help!
Best wishes,
Mike