The membership icon or flair that premium members have next to their name can be changed or removed!

First, click on 'settings' in the bottom left of the screen anywhere on Chess.com to get to your profile.

At the top of the page is your avatar, and flair, and status!

Click on your flair icon (in this example, the diamond) to change it!

The membership specific icons are only available to certain memberships. The diamond icons can only be used by Diamond members, the Platinum (crown) icons can only be used by Platinum and Diamond members, and the Gold (star) icons can be used by all premium members

All the other icons (scroll down, there are a lot!) can be used by any Premium member!