Several motivations behind these ideas, mainly:
- Could help with video making
At the moment, I know it is possible hide to all 2D HUD (cg_draw2d=0), to remove some icons (cg_drawicons=0), to remove the status bar (cg_drawstatus=0), and to completely remove all console messages (from both onscreen notify and from console itself! cl_noprint=1).
- (less important) being able not to display the chat
Setting con_notifytime "0" (instead of it default, 3, that shows console message on-screen for three seconds) shows them for just an instant. I'm sorry it's not the same thing you are asking for, and it does not affect only chat, but also system messages.
I see here http://joz3d.net/html/q3console.html
that there are some cvars related to team chat messages:
but I can't try them effectively, since for some unknown reasons (oa 0.8.5 bugs? incomplete bot chat files?), bots ignore my team commands (I write commands from say_team feature, or from ESC menu.. and they ignore me!); even enemy bots do not answer to my taunts...