I'm pretty sure that is not possible.
Hide membership level icon next to username?

I think it "can" play a role in the way someone judges and plays their opponent. Sort of like the way, if someone who has 2500 games under their belt makes a strange move, one should be very cautious on how to perceive, compared to a strange move from someone with 2 games, chances are it a mistake and not strategic or tactical. It’s really just a preference.

Chess is as much about getting into your opponents head as is it calculation. I mean I’m not super adamant about it, I was just wondering if there was an option.

I had 3 people in a row abort a 10-min game, I suspect because I paid $29 and have a gold star next to my member name. Some were rated higher and should have played.

Can I get my Modern Warfare 2 clan name next to my name?
You can put whatever name you want to in the Name field on your profile

You win--I'm terrible w/quickscope. I freaking love Rust, but it almost never shows up. :-(
Do you do FaZe quick scope vids on youtube?

This is probably the most simple CHESS.com can make more money :-):
Show the others (that do not pay) that playing the ones that pay (with membership) is a lost cause :-).
They will never let you hide this, is just way to nice feature for making more money for chess.com.
By the way I am an amateur and I've used to play without tactics and now I play lots better since I am member :-) ... they don't pay me for saying this ...
Cheerios

1. Go to your profile
2. Click on "Edit flair" (the button at the right of your nickname)
3. Select "Nothing"

1. Go to your profile
2. Click on "Edit flair" (the button at the right of your nickname)
3. Select "Nothing"
When this was posted over 5 years ago, that wasn't an option.

1. Go to your profile
2. Click on "Edit flair" (the button at the right of your nickname)
3. Select "Nothing"
When this was posted over 5 years ago, that wasn't an option.
Yeah, it wasn't

1. Go to your profile
2. Click on "Edit flair" (the button at the right of your nickname)
3. Select "Nothing"
When this was posted over 5 years ago, that wasn't an option.
Reading all the above my curiosity :::
How to know someone has premium if he even hides?
By looking at their profile.

If they hide like you told, how can I know?
You can hide the flair, not the badge on the profile.

In response to the OP - yes you can hide the diamond if you would like to do so. @CBTVET already mentioned this but here are the steps broken down in case that's helpful:
- go to your profile (chess.com/member/username)
- click on the diamond next to your name
- A dialogue box will appear for memberships badges and flair
- Select the black box with an X through it. This box is labelled with the text "nothing".
- Scroll down to add, change or remove any other flair (emojis and other icons)
- Close the dialogue box and check that your profile appears as you intended
- Play chess!
(If you don't have a membership badge displayed you can use the same steps to display it or just change your flair. Instead of clicking on the diamond next to your name, click the flair. If you have no flair displayed then you will see the box with a cross through it - click that.)
Does anyone know if it is possible to purchase an upgraded membership, however not display an icon next to your username that displays your membership status? I.e. the diamond next to ones username.