The increase in volume reduces the pressure inside the balloon. Ultimately, the pressure inside the heated balloon is probably somewhat greater than inside the balloon at room temperature, but it certainly is not as great as it would be inside a rigid container heated to the same temperature. The atmosphere, of course, is not like a rigid container. As air is heated it does indeed expand and its density does decrease. Likewise, air masses contract when they are cooled.
Warm air rises and dense, cool air sinks.