I would like some sort of message of obvious exhaustion from another player if they are trying to do something.
Why? Because sometimes when you request something from an exhausted player (a mugging, for example) they just sit there in silence because they player is trying to do the action. Sometimes they can't talk because their so tired. Sometimes they can't emote because emoting takes stamina. So they just sit there unable to really communicate their situation. Hell, even sometimes slow typers like myself try to make a show of their exhausted situation with emotes only to be shown that you're too tired to do emote it.
While we should always strive to emote and make the gameplay a little more theatrical, I can't deny that time is always an issue. The more time that passes, the more possibility that others are going to find out you're mugging and try to stop you or his buddies are going to come back him up.
The only real big con I can think that can happen from this, is a LOT of spam from the exhausted players from all the actions they attempt to make while tired.
Anyway, just an idea.