I've recently registered on chess.com to try it out. On the "My Online Chess" page there's a constant and annoying message with "Watch a Video Chess Lesson" and "Study withChess Metor".

Now, will those messages disappear if I get any kind of membership? Or just if I get the Diamond membership, which includes Chess Mentor?

Are you referring to the messages at the top of the page? Near your username? If so, they should go away with the purchase of any membership.

No, the messages on a yellow background. See below:

I don't think its very annoying, just ignore it.

Well, that's a matter of personal taste. I actually find it annoying to see some text on a highlighed background. Once you get a membership, you know where to go to upgrade. You don't need to be reminded on every page visit.