TL;DR: In most cases, being forced out implies that the employee has been coerced to quit; in this case, it probably also refers to arm-twisting of the employer by advertisers and public opinion.
Ordinarily, when someone is forced out of their job it means that they are forced in some way to resign, by an employer who wants to fire them but without the sticky consequences of an actual dismissal. For example:
Being Forced Out of Your Executive Job? 8 Things to Do Right Now
You may sense that you are being forced to quit—see the signs below that it is happening to you. Or your superiors might already have told you they want your resignation. (Lisa Rangel, Chameleon Resumes blog, September 2015)
How To Address Being “Forced Out” During a Job Interview – Ask #HR Bartender
"I recently was forced out of a position and am now job hunting. I was not fired; I left of my own volition, but it was clear my boss wanted me to go." (Sharlyn Lauby, HRBartender.com, August 2014)
US border chief 'tells agents he was forced out of job' day after Trump signs Mexico wall executive order
Border Patrol chief Mark Morgan told senior agents about his removal during a video conference on Thursday morning. He reportedly said he was asked to leave and that he decided to resign rather than fight the request. (Tom Batchelor, Independent, January 2017)
There are reasons for both employers and employees to prefer a forced resignation to a straightforward dismissal, some more licit than others; you can read more about the less-legitimate side of things under the legal term constructive discharge or constructive dismissal, as in this Wikipedia article.
However, in this case I don't believe the implication is just that Fox News asked for or forced Bill O'Reilly's resignation. Rather, the implication here is that both O'Reilly and Fox News were forced by outside actors to end their association.
The headline reads
Bill O’Reilly Is Forced Out at Fox News
This is a somewhat odd construction: if this were a traditional case of forced resignation, we would ordinarily see the word by instead of at, as Fox News would be the agent doing the forcing.
The content of the article makes this a bit clearer. From the article linked in the question (emphasis mine):
“After a thorough and careful review of the allegations, the company and Bill O’Reilly have agreed that Bill O’Reilly will not be returning to the Fox News Channel,” 21st Century Fox, the parent company of Fox News, said in a statement.
Mr. O’Reilly is departing two and a half weeks after an investigation by The New York Times revealed how Fox News and 21st Century Fox had repeatedly stood by him even as sexual harassment allegations against him mounted. . . .
Since then, more than 50 advertisers had abandoned his show, and women’s rights groups had called for his ouster. Inside the company, women expressed outrage . . . .
That put pressure on 21st Century Fox and the Murdoch family, which controls the company. . . .
In a letter to the staff Wednesday, Rupert Murdoch and his sons, James and Lachlan, the top executives at 21st Century Fox, avoided any mention of Mr. O’Reilly’s transgressions and praised him as “one of the most accomplished TV personalities in the history of cable news.”
Given this context, and the sidebar on O'Reilly's financial importance to the network1, there seems at least a strong implication that both O'Reilly and Fox/the Murdochs were "forced" against their will to end O'Reilly's contract—forced, in no small part, by the investigative reporting of the New York Times itself.
1 "Bill O’Reilly was an essential asset to Fox News. His No. 1 cable news show made about $178 million in advertising revenue in 2015, and gained viewers in the prelude to the election and since."