Can I disable global/general chat?

Solution 1:

You can remove yourself from the channel typing /leavechannel general in the chat. If you are talking in general channel, you can just type /leavechannel

Right now, however, as soon as you restart Overwatch and sometimes when you leave a match you will rejoin the General chat channel automatically and will need to leave again.

Solution 2:

You cannot remove yourself from the default global chat that you are put in while in the menus.

However, you can take yourself out the in-game chat channels by going to the Groups tab under the Social menu by hitting escape while in a game.