I haven't had a lot of experience with chase scenes in games before, mostly because they're either in a system where chases break down to a die roll, or because it's also a running gunfight and that takes center stage instead.

So the question is, how do you make a great chase scene (car, foot, horse, or whatever)?

First and foremost, there must be something exciting going on. Formula 1 is a dull activity in role-playing.
Look at Blues Brothers and Mad Max: There's a lot of action additionally to the chase itself. Obstacles into which the cars crash, people jumping from vehicle to vehicle, crashed crates, jump over ramps, drifting attempts to force the enemy into the river... That's where the very risk of dying enters the stage and the action gets evocative.

Either the system delivers me such a system (as with Savage Worlds, or, oddly if you think about the fragility of the PCs, Cthulhu), or I skip chases and hand the driving over to NPCs while the PCs do a ordinary gunfight on unstagble platforms. (I've done so in SR3 for instance - yes, there're chase rules, but they're a pain in the ass with maneuver points and what not.)