Several of the classic-era US commemorative coins include notable government buildings within their designs, here's a quick look at them - they make for a potential topical/thematic "Short Set" of US commemorative coins.
1. 1926 US Independence Sesquicentennial Gold Quarter Eagle

The 1926 Sesquicentennial Gold Quarter Eagle features Independence Hall on its reverse. Originally built as the Pennsylvania State House to house Pennsylvania's colonial government, the Philadelphia building is where colonial delegates met during the Second Continental Congress to debate US freedom which ultimately led to the Declaration of Independence. Independence Hall was also the meeting place of the Constitutional Convention which produced the US Constitution and Bill of Rights.
Today. Independence Hall is the cornerstone of Independence National Historical Park and is visited by more than 5 million people annually.
2. 1936 Columbia, SC Sesquicentennial

There are two government buildings depicted in the background on the obverse of the Columbia, SC Sesquicentennial half dollar: the building on the left (viewer's perspective) is often described as the SC State House (i.e., Capitol Building) of 1786 (though the SC General Assembly did not occupy the building until 1790), with the building on the right as the SC State House of 1936. The goddess
Justice, holding a scale, is the central figure of the obverse design.
The building is still in active use by the South Carolina Legislature, but can also be toured by the public.
3. 1936 Lynchburg, VA Sesquicentennial

The Old Lynchburg Courthouse is featured on the reverse of the 1936 Lynchburg, VA Sesquicentennial Half Dollar. The Courthouse is seen in the background, with a standing figure of Liberty seen front and center.
The building served as a courthouse from its opening in 1855 to the late 1960s/early 1970s; the Juvenile and Domestic Relations Court was the last to hold court sessions in the building.
No longer an active courthouse, today the building is home to the Lynchburg Museum, a general history museum of Lynchburg and its surrounding region.
4. 1946 Iowa Statehood Centennial

The coin depicts the Old Stone Capitol building that served as the last Capitol building of the Iowa Territory and the first of the State of Iowa. The building is located in Iowa City, IA vs. Des Moines, the current capital of Iowa; John Francis Rague was the architect for the building (he was also the architect for the Illinois Capitol in Springfield, IL.)
The Old Stone Capitol served as the seat of government for the Iowa Territory beginning in 1842 and continued in that role after the Territory became a state; it housed the Iowa Government through late 1857, at which time the state's capital (and Capitol!) was moved to Des Moines.
Following the 1857 relocation of the Iowa Government, the Old Stone Capitol was given to the University of Iowa which used it for classrooms and offices. Today, after major renovations in the 1970s and a restoration after a fire in 2001, the building remains in use as the Old Capitol Museum on the campus of the University; it is listed on the register of National Historic Landmarks.
Did I miss any?
For other of my posts about commemorative coins and medals, including more on each of the coins referenced here, see:
Commems Collection.