What is the rejected takeoff procedure for a blown tire?
With smaller planes, if you blow a tire on takeoff and have enough runway left to abort, then you should do so. Staying on the ground can be a much better option than a go-around and landing with a flat. Chop that throttle and “fly” the aircraft until it comes to full stop. That said, there will be certain issues to consider based on which wheel is the problem.
On a tricycle gear plane, popping a main will generally cause the plane to pull in that direction, and standing on the brakes aggressively can make it worse. It may require more gentle braking to maintain directional control and avoid going off the side of the runway, and this will, of course, increase stopping distance. If it’s the nose wheel that goes, directional control may be erratic and it might help to hold aft yoke to take some weight off the front gear as you apply brakes. If the plane has differential braking, you might be able to steer the plane with more authority by using stiffer pressure on one brake-side or the other.
If you’re in a taildragger, you may be straight out of luck. Keep those brake inputs punchy, and hopefully you can avoid a ground loop.
Most modern airliners have multiple tires on each landing gear strut, and the tires are designed to handle the load if one of the tires is compromised.
If the pilots know a tire has failed during takeoff at low speed, they will abort the takeoff using a prescribed procedure. At high speed, they will typically take off, perform a go-around, and return to land for a safety inspection.