Conventional light aircraft have the CG ahead of the wing centre of pressure and thus the tailplane produces a downforce to counter the nose down pitching moment of the wing. If you move the CG aft, the moment arm to the wing centre of pressure decreases, thus the need for downforce on the tailplane decreases, i.e. the stick needs to move forwards.
As the CG moves aft the static margin reduces, causing decreased static stability and making the aircraft more sensitive in pitch. Eventually, if the CG moves back far enough, the aircraft becomes statically unstable in pitch and thus much harder to control and more prone to stalling. The directional stability also decreases, making the aircraft more suceptible to dutch roll. If that's not bad enough, aircraft handling post-stall is worse for an aft CG, and the stall itself is often more violent. Other nasty things that can occur with a CG aft of limits are over-rotation on takeoff, leading to tail-stike or a stall, or PIO in pitch, particularly during the landing flare.
All in all, a bad situation. The aft CG limits is there for some very, very good reasons, and must not be exceeded. When a test crew is doing aft CG envelope expansion, it is approached VERY carefully, in an incremental fashion. The simple message is: DON'T DO IT. And if your instructor really said what you said he did, and he meant it, and you didn't misunderstand, you might want to get a different instructor....
