If aircraft tire fails during takeoff, this is dangerous in several different ways:
Flying tire fragments can inflict damage to other parts of the aircraft, as well as,other nearby aircraft or ground personnel.
Wheels with blown tires are useless for braking.Thus they greatly increase the chance of a runway overrun. It will become necessary to reject the takeoff.
A tire failure can greatly degrade directional control, especially on small aircraft.
A failed tire dragging on the runway can easily heat up enough to ignite and burn, dooming the aircraft if the crew then retract the landing gear without knowing that the landing gear is on fire.