Does the Piper PA-28 Cherokee have a limitation on its maximum lifetime flight hours?
Short answer, not in the United States; as long as the airframe is regularly maintained and passes its mechanical inspections, it can continue to fly. Some reasons this is so (when it might not be true for other models) include:


The Cherokee is non-pressurized. Pressurized aircraft are often rated for a lifetime limit of pressurization cycles (determined by the number of times the aircraft rises above and then descends below the altitude at which the pressurization system kicks in, typically 8000ft MSL for many small craft). This pressurization stresses the airframe in ways that eventually more or less requires the entire fuselage to be replaced in order for the airframe to pass its next mechanical as originally rated. The Cherokee, however, has no system for full-cabin pressurization and so its fuselage is not subjected to these stresses.
The Cherokee (180 or 235) is not rated for aerobatic maneuvers, and so does not typically perform them. Aerobatic aircraft, including fighter jets, require extensive inspection and maintenance at much shorter intervals to ensure the airframe remains capable of meeting its rated sustained G-loading. Coupled with regular testing of those ratings by the pilot, aerobatic aircraft are more quickly retired from service when it becomes impractical to repair or replace overstressed structural members.
The Cherokee is not commonly used in Part 121 or 135 operations involving passengers or air freight. As a four-seat single-engine, its capacity for passengers and cargo is limited, and so its usefulness to major airlines which are subject to Class D(econstruction) inspections every 6 years is sharply limited. Bush airlines like Era Alaska might see some use as more or less an air taxi, but even these outfits prefer somewhat larger planes for more flexibility and longer range.