The main benefit of operating with an aft CG, as mentioned, is that it can measurably reduce drag. This happens because less tail down force is required to maintain level flight in a given configuration. As a result, you can burn less fuel to maintain the same speed (or increase your speed for no additional fuel burn) compared to a forward CG.
As long as you ensure you remain within your aircraft's CG envelope, you can test this out by having a passenger shift their seat aft while in cruise.
There are, of course, some safety concerns. @yankeekilo's answer covers this thoroughly, but if nothing else it should be reiterated that an aft cg is inherently less stable.
A forward CG makes recovery from stalls much easier, and in some cases can make stalls much harder to achieve because the moment-arm of your aircraft's CG changes. This change can leave the tailplane with insufficient input during slow flight, which is why you may never reach stall in the first place; it also makes landings dangerous as there may not be enough control range to flare. If this happens, you'll wheelbarrow or porpoise the aircraft.