Is it normal for a 30 amp breaker to make a buzzing sound and to feel warm while running a drier off it? I have checked the wires and they are tight. Do I need a new breaker?

It sounds like the breaker is going bad. You should not feel heat inside the panel box. You need this changed out and unless you know what you are doing, have an electrician do this for you as this is not a good do it yourself project for beginners.

If you are able to change breaker, before you replace, remove breaker and look where breaker pushes onto Buss. Excessive current can make it buzz also.