No it will not cause any problems. You can disconnect the belt to make sure that the alternator is truly free of any internal resistance like seized bearings. That happened to me in the car once and the belt just stopped melted and broke into a billion pieces.
I am not familiar with your motor but if you decided to leave the belt off then I guess your water pump
So....assuming it free wheels .. no harm is done to the engine. To be safe you can also disconnect the wiring
in case there is some short inside the alternator that potentially screws up your electrical
, but given that it has not worked for a while and all is well, you should be fine.