Each airline publishes a document, typically called a "Contract of Carriage," that documents the legal relationship between the airline and its passengers. The Contract of Carriage may or may not overtly define the airline's policy on ticket purchases for larger passengers. In some cases, such as Southwest Airlines, the airline's policy covering larger passengers is spelled out in detail on the airline's website.
Policies vary by airline, and some airlines prefer to deal with larger passengers on a case-by-case basis. If you have questions about a particular airline's policy, it's best to email the carrier in question well in advance of booking your ticket. You will have your response in writing, which may offer you some protection if you have difficulties checking in for your flight.
Please bear in mind that airlines may change policies without providing notice to customers or issuing press releases. Policies and available information are always subject to change. It's always a good idea to print out (and read) your Contract of Carriage before booking your flight.
Current U.S. Airline Policies
News articles suggest that American Airlines will attempt to find larger customers a seat near an empty seat whenever possible. In rare cases, larger passengers may be asked to purchase a second ticket. American's Contract of Carriage vaguely refers to safety issues as reasons to deny boarding to specific passengers. (Note: I have sent an email request for information to American Airlines and will update this information as soon as I receive a reply.)
Delta Air Lines / Northwest Airlines
Delta and Northwest have merged, so the same policies apply to both. In addition, the airlines' domestic and international policies are virtually identical. For Delta and Northwest, the "litmus test" for passengers is their ability to fasten their seat belts without extenders. If passengers require seat belt extenders, they may be asked to pay for a second seat.
Southwest recently decided to fully enforce its long-standing policy on Customers of Size. As of this writing, Southwest customers who are unable to lower both armrests will be asked to buy a second (empty) seat for their flight.
United, the airline that reignited the "larger passenger" policy discussion in spring 2009, requires passengers to be able to lower both armrests and to fasten seat belts using only one seat belt extender.