While I don't doubt that the two of them had a conflict - Washington acknowledges as much in the article - I wonder if those two things are really some of the reasons the two of them reportedly didn't get along.
First of all, unless Mark is challenging Tony Dungy these days, he has to be used to hearing profanity, even from the manager. But if this is the case, guess Teixeira won't be palling around with Jets coach Rex Ryan, eh?
Second is the "smoking in the dugout" accusation. Did Washington do it? It's possible. However, as Squawker Jon noted, even famous Mets smoker Keith Hernandez went to the clubhouse to get his tobacco fix. Joe DiMaggio would smoke in the Yankee tunnel. Even Jim Leyland, the famous smoker/manager in the game, doesn't get away with smoking in the dugout anymore. There are laws at almost every stadium these days banning smoking.
At any rate, as a former smoker, I wouldn't blame Teixeira for complaining about this if it happened. I don't really like breathing tobacco smoke now (I always said I wouldn't become one of those hectoring, judgemental ex-smokers when I quit, but it does bother my lungs.) And if I were a professional athlete, I'd be peeved to be around smoking in such an enclosed space during the game.
Anyhow, in an interview with the News, Washington didn't directly address what caused friction between them, but said something happened: