Personally I find people reading the MM annoying for a few reasons, but the primary one is that it breaks the immersion of the players in the game. Instead of fighting a large green skinned humanoid monster with ropey muscles and rubbery skin they're now fighting a pile of stats.
I think this is a good articulation of the main reason that I have a negative emotional reaction to people dragging stat and/or trait info from the MM into play (whether they read it at the table or not). Nevertheless, I have told my players that they are free to access the information in the MM, and, since most in my group don't spend a lot of time on the game outside of sessions, I don't frown at them if they read it at the table. I have two reasons for this:
1) I do not want to, nor do I think I should be policing the player knowledge vs. character knowledge issue. I personally think it is more fun to "keep them separate" (although there's certainly a lot more that can be and has been said about exactly what one might/could reasonably mean by that), but I'm not going to try to control other people's thinking.
2) In order to facilitate the players making appropriately informed decisions, I feel that there needs to be a stand-in for what the PCs would know (in this case about monsters) that the players do not. The MM is highly imperfect for this purpose and I wish there were a publication that had reasonable PC monster info couched in terms that PCs might actually know. However, producing such a tome is so far down my priority list that I do not foresee myself ever getting around to it, so the MM has to do.
In any case, pretty much every time anyone pulls out the MM at the table or refers to knowledge therefrom, I issue my standard reminder that they should look at the MM as a collection of folklore - it contains a lot of good guidance, but it is frequently incomplete or incorrect about details. I think this is beginning to sink in, although one player in particular seems vulnerable to begin seduced by the notion that the MM is a rule book with the same standing as the PH (despite my reminders and repeatedly encountering evidence to the contrary).
And as for my negative visceral reaction - that just has to go on the pile of other thoughts and reactions that I choose not to act on because I think the consequences of doing so are not, on the whole, beneficial.
