When the news announced the US governmental shutdown today, it was also mentioned that tourists might be affected as well. The example that was given is that the national parks will be closed. Is that it, or will I notice other effects, such as no public transport, longer delays at the airport, etc?
Update, 2/27/15: In approximately 7 days from the writing of this edit, it is possible (Likely, in the opinion of this author), that the Department of Homeland Security will run out of money and 'shut down'. While similar to the shutdown that occurred in 2013, this current budgetary impasse does not apply to most government agencies. Parks, museums, etc., should continue to operate as usual.
The only government workers affected by this shutdown are DHS staff, such as Customs and Border Protection, TSA, and Immigration and Nationalization employees. Most of these employees are considered 'essential', and will thus be required to report to work, even in the event of a shutdown, but they will be working without pay. Expect them to be grumpy, and cut them a little slack. They may also be shorthanded, or short on supplies, as their funding has been abruptly severed. Document processing, and other services that go through an office will probably severely delayed, if not completely halted for the duration.
Again, this should not affect any national parks, museums, or tourism sites, and the FAA/Air Traffic Control are similarly unaffected. Expect delays when dealing with security, customs, and border crossings however.
You shouldn't notice much impact on the actual getting around part of travel.
Public transport, as well as things like highway maintenance and management are all done at the state level, and are unaffected by short term federal government shutdowns. Similarly, the TSA, FAA, and Border Patrol are all continuing to operate, with their employees being paid on the resumption of government activity. AMTRAK is a privately owned enterprise; while they receive federal subsidies, they should be able to continue operating at full capacity for the foreseeable future.
Where you will notice a lack of activity is in trying to use other government services. As a traveller, the primary impact you'll see here is, of course the National Parks Service. This means parks and various monuments and historic sites are closed. This includes the Smithsonian museums in Washington, DC. Do keep in mind however, that much of the nations parkland, as well as many historic sites and monuments and museums are not federally maintained or operated, and at those facilities, it should be business as usual.
While Visa processing will be delayed or affected, it won't be completely shut down. For the most part, as a visitor, you shouldn't notice much impact from the shutdown. If it drags out for more than a few weeks, the lack of available funds could cause larger issues, but for now, as a foreigner, you can safely not worry about it.
(N.B. Because of its peculiar governmental status, visitors to Washington D.C. specifically might experience a larger impact than visitors to other cities. In particular, most sites of tourist interest in DC are federally operated and will be closed. YMMV.)
For further details, CNN has a comprehensive and searchable list of which government agencies are experiencing furloughs or shutdowns, and to what degree. You can consult it if there is any particular federal agency with which you need to conduct business.