I have a 2009 S550 that makes clicking noises from the dashboard that the dealer cannot seem to fix. Does anyone have ideas on what can be causing these noises? The car has had felt tape applied numerous times. Is this problem common on S550?
Probably not common. If it were my car I would just turn up the volume on the radio but that probably will not help in this case. 
Is it a mechanical or electronic sound? Possibly radio static? Or does it happen going over bumps? Does it sound like something is loose?
What does the felt tape do? I am not familiar with it.
Yes, unfortunately it is somewhat common.
The dealer fix, as you know, is to apply felt tape in strategic places to buffer the vibration. Most of the time this quiets all rattles and unwanted noises. Obviously, in your case, they haven’t got the tape in the right place yet.
It must be really annoying to spend that kind of money and have to live with dashboard rattles until the dealer technicians can locate and silence them.
Lexus LS owners must be chuckling to themselves.
Try turning the cd player on for about 10 seconds, and then off again, and see if the noise goes away then. I’ve known some radios that the cd player makes noise if its not turned on when the car is started up.
BC.
I have been in rattling Lexus cars. All cars makes/models do it thankfully it is a really small minority. I am amazed my Subaru WRX does not given its raucous nature.
Are the clicking noises affected by the car’s speed? Bumps? Turns? RPM?
If yuo can “tie it to” some specific condition of the vehicle it should be a lot easier to quiet.