Not quite correct, actually.
The FCC has two reasons that it bans cell phones on planes.
For the first reason, the above is actually kinda close. The problem with early cell phones wasn’t so much that they would contact multiple towers simultaneously, but that they would access multiple cell systems simultaneously. The systems hadn’t been designed with that in mind, and it caused them all kinds of grief. Systems crashed, calls were dropped, that sort of thing. Over the years the systems got better, and now that sort of thing won’t happen at all. However, an airborne cell phone still ties up multiple channels on multiple towers, and the FCC doesn’t like that kind of waste of bandwidth.
The second reason is that the FAA bans them. And the reason the FAA bans them is the potential for interference on aircraft systems. Chances are an airborne phone won’t make a plane drop out of the sky. Most commonly, all it does is make annoying noises in the pilot’s headset while he’s trying to talk to the tower. Pilots aren’t idiots and they are good at their job, so they can land the plane safely even if their headsets are all going wonky. Still, it’s not a good thing to have, so the FAA doesn’t want it. Cell phones have also interfered with navigation and communication systems. But for those who think cell phones are perfectly safe, it is also believed that a cell phone caused a thrust reverser to accidentally deploy during takeoff, causing a plane to crash. Airplane systems are intentionally designed to be resistant to this type of interference, but on very rare occasions, bad things can happen.
That’s probably enough of a hijack, but worth mentioning since the above quoted text wasn’t accurate. There are plenty of other threads on the board about cell phones and planes.
Getting back on track with the OP (note - two year old thread, expect zombie jokes) the backlight suggestion by Mohamed F. El-Hewie will help conserve battery life, but in my experience the biggest drain on the battery is the phone cranking up its transmit power because it has a weak connection to the nearest tower. That part of the above posts is absolutely correct and the only way to avoid that is to turn your phone off if you’re in an area with poor connectivity, or just deal with the battery getting drained.
Basically, the cell phone is constantly talking to the tower. If it has to shout to be heard, it’s going to use up its energy a lot faster.
ETA: Cell phones (while being less than optimal with bandwidth use) do work fine on airplanes, up until about 5,000 feet or so. This is due to the directionality of cell tower antennas. There’s no use transmitting energy up into space when you can more efficiently transmit it out in the direction where a cell phone might actually be. Above 5,000 feet the chance of getting a decent connection get worse and worse, and once you get above 10,000 feet it’s almost impossible for your cell phone to get enough signal to connect. All of the 9/11 cell phone calls were made below 5,000 feet, IIRC (maybe 7,000 feet, not sure off the top of my head).