Is Snowden snowing us with his "Smurfs Suite" interview?
Yes, the government can perform nefarious acts on your phone to listen in on what your are doing. 


If you phone is truly off, there is no signal you can send to wake it up. But:


You can spoof the phone turning off. 
In some older feature phones, turning your phone off was not good enough because some things on the phone stay awake, and the FBI used this to eavesdrop on mobster phones. 


One of the best ways you can ensure your phone is truly off is to remove the battery. 
This is something they do to particular people. It's not technically feasible to do this to every possible phone, but the government doesn't have to because any given mobster or other target has a particular phone. 
This is why there is a market for undiscovered software exploits. Once you get root access, you can basically rewrite software to do what you want. 


In conclusion, this is technically possible. But that doesn't mean you shouldn't buy your friends some tin foil hats as a gag gift.