Dora did not make a mistake (necessarily) by firing Faye. Her mistake was not doing anything before firing Faye. It is not Dora's responsibility to take care of Faye, but if they really are friends then she should have at least tried something during Faye's spiral towards oblivion before it got to this point.
Dora and Faye spend a huge amount of time together (You know, the entire day, most days of the week,) Dora has no excuse for being unaware what Faye was going through. And yet, in today's comic, all Dora shows is anger.
Faye is certainly not in the right here, but I can empathize with her position. After seeing her dad commit suicide, she has major issues with abandonment. She's always struggled with alcoholism, to one extent or another using it as a painkiller when she was feeling emotional. Angus was the first solid relationship that she felt good about and had no regrets with, and he just decided to leave her because something else was more important to him. (I'll not get into whether Angus was justified here. Different argument.) Faye is crushed, and this rips open old wounds that she hasn't had to deal with in a long time. She dropped her snarky, sarcastic defenses entirely for one person, and he is now gone. Faye is in a very bad place, and one of her two remaining best friends, Marten is: One, currently dealing with his own personal life in a happy way, which Faye doesn't want to screw up by bringing in her problems. Two, someone related to her past trauma of being abandoned. (All that boyfriend drama, remember?) Both of these make him a poor candidate for helping her.
Now let's talk about Dora. Other than business troubles (Which don't seem that major seeing as the same troubles have been around since the dawn of the comic,) she doesn't really have any major conflicts going on right now, and no new relationships to be screwed up. She's Faye's oldest friend, they spend all day together, and they've both been there for each other many times in the past.
It is impossible that Dora did not know how terribly wrong things were for Faye, at least to the degree that she should have known Faye needed an intervention or a shoulder to cry into without booze being involved. When Faye says she doesn't want to spend a waking moment sober, Dora does nothing. When Faye comes in reeking of booze, suddenly chipper, and hangover free, Dora doesn't realize something is obviously wrong. She has to walk in on Faye in the act to finally react in any major way, and all we get is anger.
Based on this, Dora is either blind and stupid, or has absolutely no sympathy for her friend's suffering.
If we had seen Dora try to deal with the issue and Faye lie and say things are fine and she'd slow down the drinking, or if Dira had called Marten and Hannelore and maybe a few other people with her concerns I'd be more okay with today's strip. If Dora had only suspended Faye indefinitely, I'd be okay with today's strip. If Dora had shown any emotion on her face besides anger (Hurt, sympathy, disappointment, indecision,) for even one panel, I'd be okay with today's strip.
As it stands, though, I'd never guess that Dora was friends with Faye based off of this story arc.
I'm not saying that Dora should be enabling Faye, or letting this slide, I'm only saying that she's acting like she doesn't care about Faye as anything more than an acquaintance in spite of their years of friendship, and Faye's actions, while not justified, are at least understandable once you consider how fragile and injured she is.
Dora was justified as a boss in firing Faye, but a horrible friend in not doing anything else.