How can I remove the cover from the bottle

After years of experience of putting covers on and off, we can say with certainty that we have discovered the best method of how to handle covers. Put fingertips of both hands on the top of the cover next to the neck. Start firmly pushing down with all fingers and the same time. The first couple of tries can be a bit harder when the material needs to adjust and soften but with every try, it will get easier. 

Check the video for a better representation - for felt cover:

For feux leather cover: