Using regular gas in an engine that requires premium could void your warranty. That is most likely to happen if using regular causes severe engine knock or pinging (premature ignition of the fuel, also known as detonation) that damages the pistons or other engine parts.
For example, here is what GM says about the subject in an owner's manual for a vehicle that requires premium:
"Use premium unleaded gasoline with a posted octane rating of 91 or higher. If the octane is less than 91, you could damage the engine and may void your vehicle warranty. If heavy knocking is heard when using gasoline rated at 91 octane or higher, the engine needs service."
Note that this applies only to engines that require premium gas. Some manufacturers recommend premium gas but say that regular or mid-octane gas can be used instead. They usually warn that using lower-octane gas could reduce performance. When that happens noticeably, or if engine knock occurs, they advise to start using premium.
The computers that manage modern engines are able to adjust the ignition system to accommodate lower-octane gas — to a point. With regular gas, fuel economy and acceleration will likely deteriorate at least slightly. Because regular has lower octane, it is more prone to detonation. Burning regular in an engine designed for premium on a long-term basis or under heavy loads can cause engine knock, and that in turn can damage the pistons, valves or spark plugs. Due to the presence of knock sensors and the car's ability to retard the spark timing, you might not hear knocking, but that doesn't mean premium is unnecessary.
Regular can be used at least occasionally without repercussions in most vehicles designed for premium, but it's a bad idea to make a habit of it. Bottom line: Check the owner's manual. If the vehicle manufacturer says the engine requires premium, believe it. Don't try to save money by buying lower-octane gas.