Would a bee inside an empty bottle weights more?

If you go to school for engineering you learn about negligible forces. This is definitely a negligible force. You would need extremely precise equipment to measure the difference.

However, let's say you do have the equipment. The masses would be the same. The first bottle which sits on a scale with atmosphere inside, the second bottle sits on a scale but the bee is flying so the weight is going upwards. For an object to affect weight it's mass has to press down and the bee is suspending it's weight.

Just reading what my ol'mate Russ is saying , I would conclude that the bottle with the stationary bee would be heavier ?

It doesn't really make sense, if there was no air, the bees wouldn't fly. If you mean there's air in the jar then the jar with the bee would weigh more. Since in order to fly, the bee has to force air down which would equal it's weight. Search the bird in a truck myth on mythbusters.