No. This is one of those things that we as a people just go with. Airplane mode in phones were not even created specifically for when you are on an airplane. The function that it carries out is similar to what airplanes want, which is not having any radio interference, so they named it 'Airplane Mode'. They didn't call it that because it's purpose is to be used on an airplane.
There isn't enough interference to really cause anything. You really think that if something bad could really happen that these airlines would just leave it up to chance that people will use Airplane mode?
Nothing will happen.