Nick Berning, communications director for MoveOn.org, said his group wants to pass immigration reform legislation but doesn't support "throwing other important issues like access to health care and action to combat climate change under the bus to do it."
"We find FWD.us and Facebook founder Mark Zuckerberg's advertisements promoting the dangerous Keystone XL pipeline and criticizing Obamacare deeply disturbing," he said.
Facebook and FWD.us didn't respond to a request to comment.
The FWD.us ads are airing in the home states of senators who could be critical swing votes on immigration legislation such as Sens. Lindsey Graham (R-S.C.) and Mark Begich (D-Alaska). By bolstering their conservative credentials on other issues, the ads could give the lawmakers political cover to support immigration legislation.
Josh Orton, a spokesman for Progressives United, said that many of the groups currently have ads running on Facebook that will be pulled, but he declined to estimate how much money the boycott will cost Facebook.
"We're not out to bankrupt Facebook," he said in an email. "The point is to send a message to Zuckerberg and other executives that the grassroots won't stand for their cynical political tactics."
—Updated at 4:40 p.m. and 5:51 p.m.