When Smart in the revolving door, one can tell the glass was cut out of the upper part of each door. This was necessary to prevent glare and reflections of the studio lights and to allow them to record the spoken dialog without distortion.
When Max answers his shoe phone, he comes to attention and clicks his heels. However, he's holding one of his shoes (the phone) in his hand, so he can't click his heels.
When Max walks on screen as the group is watching the KAOS threat via the television set in the lobby, a long electrical cable is clearly visible running to Max which eventually powers the light-up waist coat button used to indicate an incoming call to his shoe phone. The cable is still present when Max is stationary in the subsequent shot.
When Smart is trying to carry on the different conversations with the three different phones, Don Adams gets the phones confused. When Smart talks to 99, he's actually talking into the phone that is connected to the KAOS agent.