Most (modern) small and large car engines are designed for 100% duty cycle. This means that at 100% rated power(gas pedal all the way down) the engine can run continuously. Heat dissipation is the limiting factor like Dave Tweed stated. Cars that are not designed to continuously dissipate 100% of the heat generated at max power require the driver to watch the temperature gauge to limit the power use.
Modern engines do not have this problem because the engine is governed (speed regulated) below the cooling capacity of the radiator. Most modern engines use electric fans on the radiators that are independent of engine rpm; greatly increasing continuous cooling capacity.
Older cars and "high performance" cars may have power that exceeds the cooling capacity. Any engine that has had the maximum engine speed regulation removed or any engine that can be "red lined" can also overheat. An engine boosting system such as nitrous oxide also exceeds the cooling capacity and thus must be used intermittently.
You will often see both large and small cars pulled over for overheating along a steep hill on a hot day. In this instance the "duty cycle" under these operating conditions was not continuous (100%). However, duty cycle is typically not used to describe this behavior, because it is a design expectation that it can operate continuously. The engine was simply operating outside of its designed range.
Duty cycle is not influenced by the size of the engine, but rather, duty cycle is a design parameter when designing an engine system. Most cars would be designed for continuous duty, while race cars would be designed for intermittent.