While the biographies of living persons policy includes a few exceptional editing powers that have been granted to prevent or reduce harm to living persons, these can be abused as some sort of trump card to give an advantage to one side in an editing dispute. This essay attempts to outline good and bad uses of invoking BLP as a justification for an edit.
"Contentious" should be narrowly construed
Looking at the history of WP:BLP, contentious material is primarily that, if untrue, would clearly cause harm to the subject. There are plenty of facts or phrasings in editorial dispute that will not rise to the level of "Mr. X is a child molester" or "Ms. Y is addicted to controlled substances". The more tenuous and disputed the actual harm should be, the less weight a BLP argument holds, and the less community support there will tend to be for using extraordinary measures (ignoring 3RR or summarily blocking an editor) to "enforce" such BLP considerations.
Facts are facts
If someone has been convicted of multiple counts of murder and grand theft, it's not a BLP violation to mention those facts with appropriate sourcing, even though most editors would agree such facts reflect poorly on the subject.
... especially to edits introduced by other editors who have extensive and collaborative edit histories. For example, drive-by editors who assert sexual orientation without support should not be extended the same benefit of the doubt that might otherwise be extended to an editor in good standing who made a similar assertion that wasn't based on an independent, reliable source. In any case, the problematic material should be removed from the article immediately.
Use the least disruptive means to solve the problem.
Notify users of the perceived issue and revert as appropriate. Explicitly state that a BLP issue applies when making a second or subsequent reversion. If an administrator, prefer page protection before blocking in most cases. If an editor inserts clearly inappropriate material in multiple places, blocking is entirely appropriate, but content disputes between editors familiar with (though perhaps having differing perspectives on) Wikipedia policies should rarely escalate to such a level.
Egregiousness must exceed involvement for involved administrator action.
It's perfectly OK to block someone for unapologetically accusing a living person of manslaughter, even if you've recently edited that same article. It's not OK to block someone for citing the Washington Post in a matter with which you disagree.
If you think it's a valid BLP issue, raise the issue without threatening an edit war
If the community, via local or global consensus, disagrees with the claim that your reversions were justified by the BLP policy's exception to the edit warring policy for the removal of deficiently sourced "contentious" material, you may be blocked for edit warring. In borderline cases, this is unlikely, as long as the community can assume good faith that you sincerely thought you were following BLP's guidance. Take special care that all prongs of the BLP policy, as currently written, are met before invoking its powers to ignore 3RR: if an objectionable statement has a reliable source, it cannot be removed repeatedly without regard to the edit warring policy.
Invoking BLP in clearly inapplicable cases has a chilling effect on discussion
Because of the importance of BLP, and the extra sanctions administrators may invoke to enforce it, citing BLP in inappropriate circumstances can be seen as a Godwin's Law type of argument, which serves to alienate and bully other editors. Editors who cry "BLP!" in an inappropriate context should be warned that such stifles free discussion, and that they may be blocked for disruptive editing if invoking BLP as justification for an edit when BLP clearly did not apply.