Mercury thermometers are not allowed on planes is because of how the mercury would react to aluminum, which is the material used for an airplane's frame. First of all, why is aluminum used?
When aluminum rusts, it produces a protective coating of aluminum oxide that protects the metal from further rusting. Also, aluminum is light, strong and cheap.
However, when mercury meets the aluminum, the mercury will break through the aluminum oxide which creates a lot of heat and then eat away the aluminum under.