Do you have the valve on the hot water line, behind the water heater, opened? The system gets pressurized on the cold water line so if the hot water valve and the bypass valve are both off the water pump will turn off once the water lines and hot water tank are pressurized. This is what may be happening:
water tank ----> water pump ----> cold water line to hot water tank ----> hot water tank ----> hot water line out ----> valve (closed) ----> hot water line to faucets ----> faucets
In summer mode the hot water and cold water valves behind the hot water tank should be on (parallel to the pipe), and the bypass valve between the hot and cold pipes should be off (perpendicular to the pipe).