What special tyres (tires) are needed for high altitude takeoff and landing?
The higher takeoff speeds might exceed the speed rating of the usual tires. As a tire is rolling it deforms in two directions. 



As each section comes in contact with the runway it is pressed inward by the weight of the plane. After it leaves the runway surface the deformation is in the opposite direction from centrifugal force. This constant flexing is what causes the tires to fatigue and possibly fail. The flexing also causes heat to build up in the rubber which causes the rubber to lose its flexibility.

The faster the tire spins the more centrifugal flexing occurs and the shorter the period between force reversals. This shortens the time the tire has to dissipate heat. 

The two ways to increase speed rating of a tire is to reduce the flexing and to reduce heat retention. The way manufacturers reduce flexing is to add cap plies. These are usually of nylon or other synthetic material placed on top of the body plies on axial ply tires and on top of the steel belts on radial tires. These help keep the body plies from deforming outward from centrifugal force. The faster the tire must spin the more cap plies are used.

To reduce heat retention the rubber used between plies and at the shoulders must be made of compounds that have lower heat retention. 

These things combined, more or stronger cap plies and better rubber compounds in key places, increase the cost of higher speed rated tires. 



Image source: The Role Of Cap Plies In Steel Belted Radial Tires [pdf], David Osborne