One question that seems to be asked a lot is if the mute button is ‘real’ on the Amazon Echo.
Lot’s of suspicion out there…. when one presses the microphone mute (circled)  it turns a reassuring ‘red’.
Question is…. does it actually do anything other than turning on a red light?
Let’s do a bit of reverse engineering to sort this question.
If one traces the signal from the push button switch it seems to flow to a small SMT part marked “ICFJ” (right most part of one’s circled below). To the left of this is a part marked “VC74” which will be a package of dual “D” flip flops and on the left the golden component is the microphone.Of course, what’s an “ICFJ”.  One of the challenges of reverse engineering modern designs is that the parts are marked with proprietary  codes.  There are a number of guides, just search for “SMD code databook” on the internet, however there is no master standard with many markings often reused.
No matter!  With a bit of acid one can etch away the package to see this die photo of the part marked “ICFJ”:
The most important bit is the text “L14D”: a sure indication that we are looking at a  74×14 class Schmitt inverter.  This inverter’s claim-to-fame is that it has hysteresis and can be used to form a de-bounce circuit for a switch.
From here a classic circuit can be sorted.  When the switch closes it can make a rapid break-make cycle which could confuse down stream digital logic.  The special property of the Schmitt gate is to remove those glitches and to provide a nice clean edge. 
This then feeds into a dual 74×74 flip-flop package as the clock input:
I then expected to find the q-bar output of the flip flop to connect back to the “D” input as this is the classic divide-by-two circuit which would provide the mute-on mute-off states… 
however the output of the flops instead go to  into this part…  another SMT mystery as it’s also marked in a non-obvious manner.  Circled here as “NEXT PART”
Of course, the nice thing about being able to de-cap silicon to extract the die is that one does not need to wonder what the part is.   
Here is the die photo of the “next part” with marking “COOR”:
Die text (M) TI2001 2G00D:       Mask copy right, Texas Instruments, 2001, 2G00D  (NAND GATE).   The output of this gate goes to what I suspect is a PMOS FET and then to the power pin of the microphones…
From here a bit more of the schematic can be teased out (simplified, some resistors not shown):
The LED is electrically tied to the signal which controls the microphone’s power…so if it’s on it appears the microphone will be off and there is no path for the processor on the soc to override without the user noticing.
The SOC (system on chip) controller seems to be connected to the flip flop resets, which makes sense as the product powers on with the microphone enabled as default…. 
Not sure why there are two flip-flops tied in parallel (one should have been sufficient?) .
But, basically if one presses the switch it causes the flip-flops to change state and to then either add or remove power from the microphones.
The  ‘mute’ button appears to be very real and functional.  When the button glows ‘red’ the power is removed from the microphones.