For recent press releases, see PR Newswire, Business Wire or Yahoo!'s Business Press Releases. You might also want to search Factiva and/or Google for the relevant company, agency, office, etc.
For older material, Dialog has two files for general press releases -- PR Newswire (File 613) and Business Wire (File 610); Westlaw's WIREPLUS database has 18 newswires (WIRES has the same 18 back just 90 days). You can also check Factiva.
Some areas with special resources include:
U.S. Government: Check the web site for the relevant agency, office or politician and/or check the general newswire sources discussed above. For older news releases, try Dialog's U.S. Newswire (File 665 for 1995 to April 1999; File 605 for May 1999 to May 2007). Also check the Web page of the relevant agency, office, etc.
Foreign: Canadian press releases are available from Canada Newswire and Industry Canada Online. Japanese press releases are in Dialog File 612.