Nothing bad happens on the plane…besides your phone's battery getting drained as it frantically tries to hop from tower to tower and fails to maintain a connection.
As Diarmaid Mac Aonghusa writes, we'd be totally screwed if something bad happened to modern aircraft just because somebody didn't switch on their airplane mode. Aircraft would have definitely suffered accidents—or intentional attacks—based on this if the electromagnetic radiation from standard cell technologies like GSM, CDMA, or LTE could interfere with modern navigation and safety systems.
The real reason that you are supposed to put your phone in "airplane mode" (from Mobile phones on aircraft):
In the United States, the Federal Communications Commission (FCC) restricts cell phone usage on aircraft in order to prevent disruption to cellular towers on the ground. [7]
Basically, having a bunch of cell phones moving along at 500-600 mi/hr, constantly trying to connect and disconnect from ground-based cell towers many miles below them interferes with cell networks. Most cell networks* weren't designed for handsets moving at such a high speed and so far above the Earth's surface. It can apparently degrade the networks significantly.
* The exceptions, of course, are cell networks that are specially designed to provide in-flight connectivity, such as Gogo Inflight Internet:
Gogo's air-to-ground network is a cellular based network that has more than 160 towers in the continental U.S., Alaska and Canada. The towers are cellphone towers that have been outfitted to point their signals at the sky rather than along the ground.
