No, as long as they are in a air tight bottle they won’t be subject to too much evaporation, or more precisely even though the evaporation process occurs, the liquid that is evaporating is trapped in the bottle and just drips back into the liquid again.
If the perfume is not air tight then it would be a slow process but could happen. It would happen faster the higher the alcohol content, so cologne would evaporate a lot faster than perfume. Even though they don’t evaporate they are still capable of going ‘off’. To avoid this keep them in a cool place in the original box or in a dark place.
Alcohol helps in binding the different components of the fragrance in addition to blending the solid components.
Now Alcohol happens to be a volatile liquid, which means it can change its state from liquid to gas if not kept under pressure externally. Therefore when you leave the bottle of Cologne or Perfume open, the fragrance can evaporate.
Credit: https://www.quora.com/