First, let's talk about the basics: size and weight. If your hair dryer is a compact, travel-sized one, you're probably good to go. But, if you've got a full-sized hair dryer, you might want to check with your airline first, because it might be considered a checked luggage item.
Now, I know what you're thinking: "But I need my hair dryer to look fabulous on my trip!" And, trust me, I get it. A good hair day can make all the difference when you're traveling. But, the TSA has to consider the safety of all passengers, so they have to draw the line somewhere. And, that somewhere is usually at lithium batteries and heat-producing devices.