About one in eight employees cite their employer’s mission as the main reason they stay on the job, according to a recent survey of 36,348 workers by Comparably, a workplace culture and salary site. Companies often embrace lofty mission statements partly to retain workers, such as vowing “to inspire humanity” ( JetBlue ) or “to refresh the world in mind, body and spirit” ( Coca-Cola ).

However, intense pursuit of a mission can foster groupthink and resistance to change. Companies need naysayers to bring problems to light—even though speaking up can jeopardize their standing in the organization, or even their job.

Dissenters face these conflicts at companies and nonprofits alike. Joanne Sonenshine was dismayed when her former employer, a nonprofit promoting sustainability, began shifting funding toward courting and entertaining corporate partners, away from the grass-roots development work she valued. The organization’s mission had been a major reason she signed on. “I felt strongly I was doing my part to change the world,” says the Arlington, Va., economist and author.

She says she tried to persuade leaders to shift their focus, but faced mounting attempts to micromanage her work. She eventually lost hope. “I walked into my boss’s office and said, ‘I’m totally done,’ ” she says. She has since founded a consulting firm, Connective Impact, to help clients set philanthropic priorities and choose projects that will have the greatest impact.

Inspiring mission statements can blind people, says Adam Grant, author of “Originals,” a book on nonconformists. “The point of a mission is that it’s never finished, or never entirely achieved. This can make people extremely zealous about doing whatever it takes to fulfill it, even if some of their actions are at odds with the organization’s values,” says Dr. Grant, a professor of management at the Wharton School.

Some employees mount protests about ethical issues via Coworker.org, a meeting place for dissatisfied workers. Dissenters are most vocal in Silicon Valley, home to some of the most ambitious corporate missions. (YouTube aims “to give everyone a voice and show them the world.” Snap chat’s parent, Snap Inc., claims to “contribute to human progress by empowering people to express themselves.”)

Former employees and managers have formed or joined advocacy groups to battle what they say are the addictive, divisive effects of the technology they built. Many signed petitions recently protesting their companies’ work with Immigration and Customs Enforcement over the agency’s role in separating migrant children from their parents.

Sandy Parakilas, a former platform operations manager at Facebook , was drawn to the company in 2011 partly because of its mission of connecting people and its success in doing so during the Arab Spring uprising.

He says he tried while working there in 2011 and 2012 to call executives’ attention to data-security problems, sending them a slide presentation on the risks. Facebook’s data-privacy problems hadn’t yet surfaced publicly at that time, and Mr. Parakilas got what he calls “a feeble response.” He left the company and has penned op-ed articles criticizing the company.

Facebook changed its mission statement last year to put more emphasis on building community. Its previous mission was “to give people the power to share and make the world more open and connected.” Now it aspires to “give people the power to build community and bring the world closer together.”

Facebook encourages dissenters to speak up, says Bertie Thomson, director of corporate communications. “Some of our hardest product and business decisions have been the subject of very heated debate” internally, she says. Facebook has said in an internal posting in response to Mr. Parakilas’s criticism that while it was fair to criticize its enforcement of privacy policies in 2012, it’s devoting hundreds of people and new technology to better enforcement now.

As chief strategy officer at the Center for Humane Technology, a nonprofit launched in February, Mr. Parakilas is advocating now for technology and policies to make using social media safer, less addictive and less vulnerable to manipulation.

People who object on principle to their employers’ conduct face many obstacles. One is the bystander effect—people’s reluctance to intervene against wrongdoing when others are present and witnessing it too, Dr. Grant says. Ask yourself in such cases, “If no one acted here, what would be the consequences?” he says. While most people think first about potential damage to their reputation and relationships, the long-term effects could be worse, he says.

Be careful not to argue too passionately for the changes you want, Dr. Grant says. Show respect for others’ viewpoint, and acknowledge the flaws in your argument to show you’ve thought it through carefully.

Be open about your concerns, says Jonah Sachs, an Oakland, Calif., speaker and author of “Unsafe Thinking,” a book on creative risk-taking. People who complain in secret are more likely to make enemies and be seen as disloyal, compared with those who resist in the open, research shows.

Successful change-makers tend to frame proposed changes as benefiting the entire company and its employees and customers, rather than just themselves, Mr. Sachs says. He cites a former executive at a retail drug chain who helped persuade top management to stop selling cigarettes in its stores. While the move tracked with the company’s health-focused mission, the executive strengthened her case by correctly predicting that it would attract more health-minded customers.

Sydney Cavero-Egúsquiza, a dietitian and nutritionist, became uneasy when a former employer asked her to help run a weight-loss competition in its retail stores. She’d seen so many clients discouraged by repeated dieting failures that she had begun counseling a gentler approach. Ms. Cavero-Egúsquiza of Tulsa, Okla., says she pushed back diplomatically, urging her managers to run a healthy-living program and prize drawing instead, to no avail.

She was soon laid off in a downsizing. Ms. Cavero-Egúsquiza has since started her own nutrition-therapy firm, Simply You Nutrition. She says she has no regrets about voicing dissent. “I never have to look back and think, I wish I’d spoken up,” she says.

Write to Sue Shellenbarger at sue.shellenbarger@wsj.com