The phrase "to force any resignations" means to make someone quit their job or position against their will by putting pressure or creating conditions that make it difficult for them to continue. Full definition
They did force the resignation of the 3 people in the medical profession who supported our decision. (biggerpockets.com)
The legislators say they have no authority to force the resignation of the board but wanted answers. (spectrumlocalnews.com)
But on Tuesday, Huma Abedin was front and center as Weiner confessed to having further explicit exchanges, even after the first scandal forced his resignation from Congress. (politicalticker.blogs.cnn.com)