One of my favorite examples: Lego Island's driving mechanic is tied to frame rate, so on a modern machine tapping left or right will fling your car 90+ degrees: https://www.youtube.com/watch?v=2CmqbccCqI0
I discovered this playing the old DOS RPG Drakkhen, where there was a tower with a shark in its moat, circling around. It would jump and eat your characters if they happened to cross the bridge at the wrong moment. The funny thing is its speed was based on the processor clock (the game was released in 1989) and if you played the game with a faster processor, the shark would be too fast to be avoided. Push the turbo button, reduce clock speed, problem solved.