Would there be any benefit in placing the aircraft engines on the elevator (end-tips)?
There are some issues with placing engines in the horizontal stabilizer tips (I'm assuming you mean putting the engines in stabilizer- elevators are the control surfaces):


The engines are heavy- putting them at the rear shifts the c.g aft. The associated change in the c.g requires a larger horizontal stabilizer.
Putting the engines in horizontal stabilizer tips mean that it has to be strengthened, increasing the weight
Also, the fuel lines etc. has to be routed through the elevator, which increases complexity and weight penalty (not to forget the control surfaces which have to be there, which places a lower limit on the horizontal stabilizer size).
Not only the engine weight has to considered, but also the fact that the thrust has to be transferred from the engine, which furthers the weight penalty.
Noise in the rear of the aircraft- though this can be reduced by placing them away from fuselage, this further increases the weight.
Another thing to consider is that the engine may be affected by the wing wake, especially at high angle of attack.