Passengers inside a steel bodied car are protected from lightning because the car behaves like a 'Faraday Cage'. When lighting hits the car it wants to take the shortest path with the least resistance to ground. That is via the body shell and not via the occupants.
This was demonstrated several years ago on the BBC entertainment show 'Top Gear' where they put a hamster inside a car:
As for the tyres, made of rubber, which does not conduct electricity?
Well, the tyres are made of vulcanized rubber and the black stuff in them is carbon that does conduct electricity, as do the steel belts inside the tyre. The question is how much does this conducting-electricity-thing amount to?
Pneumatic tires which contain carbon black generally have a resistance of approximately 10^6 ohms, measured from the crown to the bead. There is a provisional specification for this measurement, this being laid down in the WDK guideline 110. According to this a pneumatic tire belongs to "electrostatically active Class I" if it has a leakage resistance of less than 10^6 ohms.
Pneumatic tires which do not contain carbon black do not satisfy this test; their electrical leakage resistance is approximately 10^10 ohms.
How does that compare to plain old air, as in that stuff we breath?
Generally we do not use air or any other gasses to conduct electricity and, for the sums, air is considered to be of infinite resistance. But, where there is a will there is a way and, if you place a big enough potential difference across air it will conduct. When this happens the little electrons in the air get scared and do a runner from the nuclei that they have been bonded to in a process called ionization. All of a sudden the air conducts.
So what does Mr Lightning make of this? Well, he knows he can ionize air as he has already done quite a lot of that already making his way through loads of sky. However, as he gets to the ground he wants to be lazy and discharge his electrical load as easily as he can. Spotting a car in his path he thinks 'I will go and hit that rather than the nearby ground as that is closer and less effort'. Then he gets to the bottom of the car and sees some more of that air stuff. Not wanting to strip all those air molecules apart again he cheats and goes for the tyres. The carbon-black tyres are planted on the ground and, although not exactly as easy a ride as copper would be, he does the maths (10000000 ohms resistance is better than infinite) so he goes that way.
Being a cheeky so-and-so Mr Lightning would dearly like to zap the hamster in the car, however, no matter how hard he tries he just cannot ionize the air inside the car in the way that he has been used to to so far. Not because the air conditioning is off, but he cannot get the required potential difference, what with that tin box conducting so readily.
So, to answer the question. The tyres play no role in protecting the occupant from lightning strikes. This is not because they are insulators, but because they conduct electricity instead. They have to conduct electricity to prevent static build-up in the car and there are standards in place to make sure that they do not have too high a resistance, even if they do not have 'carbon black' used in their manufacture.
It is the Faraday cage of the car that protects the occupant, a wrap-around lightning conductor that works very well so long as it is made of a metal.