After Neo and Trinity rescue Morpheus, they are all in the subway. Right after Morpheus picks up the phone and is sent back Neo puts the reciever back on the hook. Listen! You can here the coin being deposited. You only need to put a coin in the phone if you are making a call, not recieving one.
I think that it is just the phone being picked up off the ringer, because if you notice, public phones make a noise when you pick them up. Just thought I might add that.
I'm actually from Sydney Australia (where the matrix was filmed), and i can tell you that that is correct....the pay phones here do make that sort of 'click' noise when the receiver is picked up or hung up.....a similar sound to that of a coin dispensing.