Switching it off will not break anything. Cold tap in the kitchen is direct from the mains, you may also find that the toilets are not filled using the pump (easy enough to check, switch it off and flush the toilet, if the cistern refills as normal then you know it is fine). Given you haven't found any leak, it is likely that the diaphragm is degraded and that is causing the pump to switch on. You may still get a flow to various taps (especially as the tank is upstairs from what you are saying) but it will be low pressure, certainly not enough for showers or anything.