What happens to the weight of water during heating?

Nothing. In a closed system you cannot create or destroy mass by heating. Picture taking a sealed cansiter of water and heating it on the stove. The water inside would get hotter and the pressure would increase and it would all eventually become steam. But since no water has escaped and none has been added the amount of mass inside the cansiter hasn't changed so it will still weigh the same. The only time it would change is if you had an open system, like an uncovered pot on the stove where the steam is allowed to escape. Since steam is exiting the system the mass is decreased and the weight is decreased.