It is unlikely that the hair dryer overheated any of the surface mount components. However it may have caused some solder to flow to an area where it should not have, possible shorting things out or preventing proper function. Low temp solder can be heated to this point by a hair dryer.

The actual "beeps" do NOT originate from the ESC at all!

Rather the ESC pulses the motor windings to produce the tones you hear.

If the motor is disconnected or improperly hooked up, you will hear NO tones from an otherwise working ESC.

Also if the ESC is not seeing a signal from the receiver it will act much as you've described.

Check that your throttle channel is working properly by hooking up a servo to the throttle channel.

Also you may want to perform a continuity test from the throttle RX plug back to a surface trace wire on the ESC itself.... of course this is done with everything unpowered!