Could repeated running of a hydraulic system cause overheating of hydraulic fluid? if so what would be the possible implications of this?
As long as the design of the aircraft is respected on the ground not common at all, in flight not a chance given a properly working aircraft. As said above some aircraft have heat exchangers in the fuel tanks requiring a minimum amount of fuel for cooling most just rely on the hyd lines expelling heat and at altitude it is cold enough out where even the aircraft with fuel heat exchangers do not really need them.
Now the most common cause of an overheat is due to an actuator failure where fluid that is normally contained on either side of a piston slips by at super high speeds thus friction thus heat.
All aircraft I know of have a means of monotoring hyd temp, if you are concerned about long term usage on the ground just keep an eye on the temperature and give her a break if need be.
Implications.
The pumps and seals take the highest toll soo pump failure, component failure or leakage is possible.