As a general statement aircraft are pretty well-built (in fact, they are generally overbuilt so they'll hold up to use and abuse in excess of their stated design tolerances), and carefully maintained. As such there is generally no regulatory restriction on airframe age, though there may be other applicable restrictions (or recommendations) depending on the aircraft.
The most common restriction you'll encounter that's similar to what you're asking about is what are known as "life-limited parts" - Aircraft components that have a finite limit (expressed in flight hours, flight cycles, or calendar time), after which they must be replaced or overhauled.
These limitations are usually given by the manufacturer (in a Limitations section of the POH/AFM or Maintenance Manual, which has regulatory effect under FAA rules).
In some cases an entire airframe may be life-limited: (The R22 helicopter has both a calendar and hours limit, and commercial airliners typically have a flight cycle and flight hour limitation on the airframe as well.
In other cases only a component may be life-limited: The wings on a Grumman Tiger or a Piper Tomahawk must be replaced after a certain number of flight hours.
Even in absence of a specific limitation it's important to recognize that all airframes have a finite life - components will eventually suffer fatigue and fail, as with any kind of mechanical equipment. Manufacturers may recommend extensive inspections of older aircraft to ensure that fatigue is detected before the structure fails. (For example, while the Piper Cherokee family has no Limitations published Piper does recommend extensive inspections on aircraft that have accumulated a large amount of flight time.)