"The angry can not find me, yet the proud goes to my door."
Angry people tend not to listen reasoning but when they're proud of something they have achieved they want to hear praises.
"My tail eludes me, when through the wind I soar."
Probably a reference to The Doppler Effect, eh?
"My shrines are east and west, my drums alert my master."
Shrines and the cardinal points refer to ears on the sides of one's head. Eardrums transmit the sound to the inner ear eventually the brain gets alerted.
"But give me a dollar value, and you'll lose me even faster."
One dollar is not much. If you do not value your hearing high enough and protect your ears during loud activities then you will definitely lose it.
Thus I believe the riddle pertains to:
hearing.
loose
? Or should it belose
? $\endgroup$