I can see Burnham having a soft spot for Emperor Georgiou because she sees traits in her that remind her of her captain. I can see Georgiou having some personal feelings for this Burnham, too. After all, at least this one hasn't betrayed her, and it doesn't seem like she will. Of course, Burnham also has to remember she saw this Georgiou eat Kelpian and she did destroy a rebel outpost in cold blood.
That said, I can also see her wanting to be protective of Georgiou against Lorca, given what she's found out about him. But however this shakes out, I think things are going to come down to Burnham having to choose between her feelings for Georgiou and Lorca and work to save one and destroy the other. Thing is she may find it's Lorca who is really the one on the right side of the conflict within the Empire. Not necessarily to bring it down, but at least to be a less ruthless Empire under him -- or he may even want to declare a republic. Lorca may be our hero, yet.