How come car and bike tires don't catch fire even when driving on a hot road in hot weather at high speed or even when braking suddenly?
Basically two reasons : the material is designed to support the temperatures in normal use (excessive temperatures can be caused by low pressure and subsequent failure) and the tyre also radiates heat to the atmosphere / surroundings - you can feel the heat being radiated from warm tyres if you put your hand close after a fair distance.
It is possible to cause a tyre to catch fire - check out some of the "burnouts" of bikes and cars, but this is not "normal" use.
The elegantly simple answer is; if they did, we wouldn't use them. If they designed a tyre that set itself on fire, they wouldn't pass type approval and they wouldn't allow them to be sold.
The rubber compounds used within the tyre are chemically engineered to prevent this from ever happening.
I have once seen a tyre catch on fire. It wasn't the actual tyre itself, it was the "marbles" (debris) thrown off the tyre getting trapped in the end of the sill (rocker) panel and sitting on the hot exhaust. It was on a race car, at Silverstone race circuit, during a 24hr endurance race on a particularly hot day and the tyre in question was made of a very soft compound, a Dunlop racing slick. The temperatures involved were incredible (the exhaust of a Viper GTS-R at race speed) and not something you'd be likely to encounter on the road. The volume of rubber that had gathered on the exhaust was a good few hand-fulls and it had been accumulating for some time before it ignited. Also, it was very easily extinguished.
According to Wikipedia, flame retardants are added to tires during manufacturing which prevents tires made of carbon from catching fire. Source: en.wikipedia.org/wiki/Waste_tires
In normal driving conditions tyres are only subject to rolling friction in the forward direction and static friction when accelerating, braking and cornering. Unlike say brakes which are operating under sliding friction this doesn't generate a great deal of heat.
Indeed most of the heat a tyre does generate comes from deformation of the tread block and tyre carcass under load rather than friction as such.
If you lock the wheels under braking, spin them under acceleration or slide in cornering then they can get hot enough to start to melt and break down producing smoke and laying black lines on the road surface.
But even then this is quite localised and the body of the tyre isn't going to get hot enough to catch fire. F1 tyres for example, routinely operate well above 100 degrees C, in fact they need to be this hot to work properly. An overheated tyre is more likely to fail mechanically before it catches fire.