Front tire will cause sloppy steering and much reduced traction. Kind of like driving on ice. Over the short term, the tire will disintegrate. Depending on how this happens, you could get part of the belt fouling in the suspension and causing considerable body damage as things start tearing apart. You're also going to lose ground clearance, so obstacles in the roadway that would normally not be a problem could catch the frame or suspension and stop you.
Eventually (the one time I had to do this, it took about 30 miles of mountain road) the tire will be pretty much gone and you'll be rolling on a pair of steel rims-- at least until they flatten out. This will cause quite a bit of heat, so look for your brakes to start getting mushy, or possibly seizing if the rotor warps (this is era specific, so if the story is a period piece in the '70's, for instance, you'd have different brake issues because of the different types of brakes used back then). 1 brake locks up and if you aren't ready for it, or don't know what to do about it, you stand a good chance of spinning the car.
Due to the lack of shock absorption the tire would have provided, you might lose some lugs (those threaded bits you use to connect the wheel to the suspension) You lose the wheel and you're driving on the drum/rotor, and THAT'S a party. Even less ground clearance, if any at all, and driving on a part that's not designed to hold weight. You could easily end up with seized calipers (the bits that slow and stop the rotor. They work like rim brakes on a bike) or a warped drum (older cars) Rougher, harder roads will make this happen quicker.
Rear tires. If you don't have positraction, where both wheels are supplied with power equally, generally what happens is that power is applied to the wheel experiencing the least resistance. That would be the one with the flat once the bead breaks and the rim is spinning freely inside the tire, or when the tire falls off. This means you probably aren't going anywhere. In this case, you'd almost be better off if the brake did lock up. At that point, you'd be dragging that wheel, and the noise you wouldn't believe, but at least the other wheel would now be providing traction (I actually serviced a car where the woman who owned it dragged a frozen brake until her brand new tire exploded, changed to the spare, and dragged THAT one until it exploded. I cannot imagine how she didn't hear that squeal.)
If you DO have positraction/limited slip, both wheels provide power regardless of drag. You'll get some slew as the tire flops around on the rim. This could break things like brake lines or body work. Very hard on the chrome. You'll be getting lots of stress inside the differential, but in the time frame you're dealing with, it won't be an issue.
Finally, with any sort of flat running at high speed, you're going to see extreme heat. The least you can expect is to see globs of melted rubber slinging everywhere. At worst, the tire will catch fire. At that point, brake fluid starts to expand, and that's flammable too. You've also got fuel lines running along the frame on most cars, and you do not want open flames (and rubber burns hot) around those, as many of them are only steel until they bridge the gap between frame and engine. Then there's a rubber(ish) section that melts under intense heat. The smoke is pretty noxious as well, and might be a greater hazard than the flat itself.