Light propeller aircraft usually do, but commercial jets aircraft do not.
The start sequence for a jet aircraft typically involves firing up the auxiliary power unit, usually located in the tail of the aircraft, which feeds compressed air that spins the engine and, when a certain rpm (revolutions per minute) is achieved, activates the engine's igniters, which causes the fuel to burn. That's a matter of flicking switches in the right sequence and at the right time, but no ignition keys are involved.
Despite the lack of a key, getting into a cockpit is not easy. Commercial pilots are checked at several stages before they can enter the cockpit, usually with identity cards which must be swiped at key points.
While it might be possible to incapacitate the pilots, steal their ID cards and enter the cockpit, that's Hollywood. In 2013 a French national wearing a pilot's uniform conned his way into the jump seat in the cockpit of a US Airways domestic flight. When he appeared not to know how to buckle the seat's chest-harness belt, the pilots smelled a rat and called in the cavalry.