So there are a lot of people on here posting (at best) conjecture.
I am an airline pilot with 16 years professional experience and in excess of 10,000 hours (mostly at the helm of a 747).
I can categorically say that mobiles CAN and DO interfere with aircraft systems.
In my personal experience, it has been limited to radio transmissions and reception. The interference sounds exactly the same as holding your phone up to an unshielded speaker.
I have only ever noticed this whilst flying smaller, older aircraft. I'm guessing that it was more noticeable due to the proximity of 'phone to aircraft RF antenna, and also due to the older aircraft RF shielding being of a lower standard. Personally, I have never experienced any interference with aircraft navigation/non RF systems though. However, that is not to say that it doesn't/couldn't occur.
To be honest though, radio (comms) interference is bad enough - the pilots could easily miss a vital instruction/readback from ATC - at high OR low altitude.
There is (correctly) some talk of phones and devices being TOTALLY off for take off and landing. This is, as suggested by some, to ensure that passengers are distracted as little as possible and therefore stand the greatest chance of surviving an evacuation should the worst occur - the vast majority of aircraft accidents occur during the take off and landing phase. This has nothing to do with interference and everything to do best practice - aviation is the safest form of transport in the world for a reason; tried and tested procedures and attention to detail.
As a sub-note, some airlines are introducing low power base stations/femtocells onboard aircraft that connect to ground cell networks via satellite. These are, of course, fine as the aircraft and systems in question have been checked and correctly shielded.
Newer aircraft = less likely to be a problem
Follow advice of the crew.
If in doubt, keep your phone in flight mode.