Here's what I think is really going on with Hartley Sawyer's apparent firing: I think it's fake. I know how that sounds, so please let me explain myself. By pretending to fire him, it draws much more attention to the issues of racism and misogyny compared to if they stood by the person he is now because he is a very popular character on the show. I say this because there were many inconsistencies with Hartley's firing when compared to other people who have been in Hartley's shows:
1. Roseanne Barr's show was canceled the same day she made the Muslim and Planet of the Apes tweet. James Gunn was fired the same or the next day after his tweets resurfaced. Hartley however was fired more than a week after his tweets resurfaced, based on the date of his apologetic Instagram post on May 30th and he was fired on June 8th.
2. Not one of Hartley's costars have voiced their support of him whereas those who know Hartley personally from outside the show have. However, his Flash costars still follow him on Instagram. If they really condemned him for the person he is now, they would have unfollowed him as Danielle Panabaker did to Candice Patton for whatever reason. This suggests this was all planned by the cast and crew of The Flash. It also explains why they haven't voice their support for him like the cast of Guardians of the Galaxy did for James Gunn.
This tells me they either plan to come out with the truth about Hartley's firing and hire him back outright. Or they plan to let him come back down the road and pretend to give him another chance. I say "pretend" because they never really fired him.