If you think about what a steel belted radial that also possibly has Kevlar reinforcment in it does day in and day out, namely, hold up a multi-thousand-pound car at 75 MPH while rubbing on rough pavment, you'll realize that tires in motion are actualy rather bullet resistant, especialy to handgun calibers.
When you factor in the tire's rotation on a moving car, and presentation angles far removed from a perfect 90 degree impact, and that the rubber has a fair amount of give and flex, not to mention the layers of steel and aramid fiber reinforcments, a tire is a very tough thing for a handgun round to bite into.
And none of this addresses the difficulty in hitting a moving target, or worse, a moving target from a moving platform, such as a persuing car.
A centerfire rifle with a spitzer profile bullet is another story, however, you still have riccochet problems in public spaces and liability concerns.
Shooting tires is a desperation move at best. I would think that LEO's have to have good reason to try it, mainly concrete cause to say that the fleeing felon is a greater risk to the public than shooting at tires.
As a civilian I'd only risk it if a loved one was being taken in the car, and I believed there was a real threat of rape or murder.