Umbridge ordered Trelawney out of the castle, but Dumbledore appeared and countermanded that order. Although the High Inquisitor had the right to fire teachers, as Headmaster, Dumbledore still had the authority to determine whether or not she could live at the school, and he wanted her to stay. I was interested to note that McGonagall, who is normally about as far from being an admirer of Trelawney as it's possible to get (when I had to explain to her that I was dropping Divination, she gave me the sweetest smile!) was right there, showing Trelawney that she was on her side. So was Flitwick, and, scannning the crowd, I don't think that any of the teachers disagreed with them. Even Snape looked deeply disapproving of the whole mess, although I think he was at least as revolted by Trelawney's public breakdown as I was. That should have been a big red flag for Umbridge, since almost all the teachers think that Trelawney's a fraud and show-off.