I boo'd a lot back in the Behring/Flores days and don't feel a bit bad about it. I paid good money to attend games in which it was patently obvious my team didn't care. They deserved the boos.
It's none of Tate's business if the fans choose to boo or not. They paid for their tickets, and lots have season tickets they owned before Tate was ever a Lion. Tate is an employee, the fans booing are the customers. He should be grateful they actually bought the tickets and showed up, since without them, he wouldn't be playing a sport for millions.
And I don't get fans telling other fans how to behave. As long as they're not being violent or disruptive, it's none of your business how they choose to enjoy the game. Nobody is telling the critics they have to boo. So why go tell others they can't?
Frankly, sometimes these teams need to be booed so ownership will wake up and make the appropriate changes.