Hearing an echo of my voice using the Apple headset
I am having the same exact issue. I have called Apple Care and all they could tell me was to do a soft reset, and a full restore and that the cause would be hardware related. I don't believe it's hardware related since this occurred after the upgrade to iOS 6 so this is clearly a software related problem. They made no other suggestions on how to resolve this issue at this time. 

I did however google this and found this post.

This seems to have been an issue even with iOS 5 on the 4S. This only happens when plugging the headset in during a call. If you initiate the call with the headset already plugged in the echo is not there. If you plug the headset in during a call however, in order to get rid of the echo, turn the speaker on and off (with the headset still plugged in) and the echo will go away.