Ever heard of the verb "to dooce"? From today's Chicago Tribune:

First there was Heather Armstrong, the constipated former Mormon Web designer who lost her job when her bosses saw what she'd been writing about them on her Web site. Then there was Joe Gordon, sacked from his job at Britain's biggest bookseller, Waterstone's, after grousing on his blog about life at "Bastardstone's." Then there was the Phantom Professor, unmasked and unemployed after writing about her spoiled students and supercilious tenured colleagues at Southern Methodist University.

And now there's Jan Pronk, the United Nations chief envoy to Sudan, sent packing after blogging about military setbacks suffered by Sudanese troops fighting rebels in Darfur.
...
Getting canned for something you wrote on your Web site is such old news that there's a word for it--dooced, which comes from Armstrong's site, dooce.com. But this has to be the first time a diplomat has been dooced.

I could have sworn we've talked about this before, but I couldn't find it. Is it sometimes called something else?

Ah...here it is.

I know of one person at my own library whose position was made more unstable because of blogging, but I think that person quit (with encouragement). What words are there to describe that sort of "encouraged" quitting?

The phrase normally used here is that they left 'by mutual consent'.