Many sites in D.C. require tickets for entry. Some require them year-round and some require them only during the tourist season, primarily March - August. Most of these sites offer the opportunity to acquire tickets in advance for a small booking fee.
It is also sometimes possible to obtain tickets to some of these attractions through the office of your U.S. Representative or U.S. Senator.
Foreign visitors can obtain tickets online or in person to any of these sites except for tours of the White House. Foreign visitors would need to go through their Embassy to request tickets and many Embassies do not handle these requests. Therefore, their citizens cannot go on tours of the White House.
1. White House Tours
http://www.whitehouse.gov/about/tours...
2. U.S. Capitol Tours
http://www.visitthecapitol.gov/
3. Washington Monument (Access to the top via elevator) (Indefinantly closed because of a recent earthquake)
nps.gov/wamo/…feesandreservations.htm
4. Ford's Theatre (Various combinations of access to the theatre, the museum and ranger talks)
http://www.fordstheatre.org/ and click "Plan Your Visit"
5. Holocaust Museum Permanent Exhibition (from March - August)
6. National Archives (Access to the Archives does not require a ticket, but the Timed Entry is recommended in the tourist season due to long lines. Guided tours are also available.)
www.archives.gov/nae/visit/ and go to "Plan Your Visit"
7. Bureau of Printing and Engraving (The BPE requires tickets for admission from March through August but these are only available on a same-day basis)
http://moneyfactory.gov/tours.html
Finally, many of the "pay" museums offer ticketing in advance. Of those, only consider getting tickets in advance (during the tourist season) for the International Spy Museum because those tickets are time-specific; some times do get sold out; and because the lines to purchase tickets once you get here can be quite long.