Your teacher weighs a bottle of perfume and then passes it around the whole class so you can smell the perfume. Then your teacher weighs the bottle...

The primary reason is that the amount of mass lost by a room of people sniffing the perfume is basically negligible. Yes, particles are traveling from the perfume bottle into people's noses, which could take some mass away, but that potential loss of mass is so small that you would need extremely sensitive equipment to detect it.

There is another reason, as well; if the perfume is not pressurized above atmospheric pressure, then when opened in the air the amount of perfume that flows out will be the same as the amount of air that flows in. If the perfume had approximately the same density as air, then the bottle could go from completely full to completely empty without getting substantially lighter.

If it had been a pressurized container, then it would actually get lighter when opened, because more gas would flow out than flows in. Try lifting full and empty propane tanks sometime. Because the propane in a full tank is pressurized, the tank contains more mass when full than it will once the pressure has equalized and the tank is empty.