my geberit was leaking on and off for the last 2 years - but only after the cistern had filled. recently it just wouldn't stop running. so i called 2 plumbers - one had no idea how to proceed and ran away, and the other who had experience with concealed systems recommended to rip out the wall to gain access to the cistern at a starting price of £320 (don't do this! the cistern is sealed, and the only access is through the 4x6 opening). unless you have a Geberit specialist on hand, this is job you need to do yourself, IMO.
i had been reluctant to unscrew the front plate and start fiddling around with the mechanisms because i feared an out-of-control flood onto the flat below. don't worry about that. the shutoff valve is a blue knob on the left once you get the plate off. turn it off counter clockwise and then flush the toilet.
the plate on my system had visual instruction on troubleshooting what i suppose is the most common problem - running. i didn't know what it all meant at first, but when i got into the guts of the thing and popped the center bit off, i could see what was going on.
to the right is the post that regulates the water flow, knowing when to shut itself off. over a few years, this mechanism can get gunked up, and needs to be flushed under a faucet. but you need to take it off first to clean it, and that's a bit of a struggle. the instructions show you how. you pop it off the screw device on one end and twist it on the other to remove it. don't drop it! i feared this, but it didn't happen. if it happens, just fill the cistern and it should float to the top.
clean it off really good under a faucet, then reattach it - again, a struggle. make sure the sleeve is aligned properly, or the water won't flow when you turn it on again (happened to me). i did all this and everything seemed fixed, then it started leaking worse than ever!
solved: there is another adjustment mechanism which is that screw bit that the end of the blue cap attaches to. by turning the screw, this can to be adjusted up or down to let the system know when to turn off - sort of like the floating ball in a regular toilet. mine was adjusted too high - at about 10 notches above the plastic base. i dropped that down to 20 notches, and that solved the problem. in fact, adjusting this screw mechanism should be the first thing you do in troubleshooting this type of problem.