When someone tells us what we should/should not do, we need to bump their instruction against The Word which, for all my looking, doesn’t specifically tell us we can’t use any particular hashtag, regardless of what it is.
What The W0rd does say is that Christians are to be slow to speak, slow to become angry, and to mind a tongue that is a fire and a world of evil among the parts of the body (James 3:6), we all know that. So, if we also know a hashtag will cause anger and stir up conflict then we have good biblical reason to refrain from using it. However, the idea that #alllivesmatter should be avoided because it’s offensive comes from culture and culture is a terrible guide for any Christian.
Culture tends to think telling people they are sinners is offensive. Should we stop doing that? Culture thinks Hell is offensive. Should we stop talking about that? What about biblical marriage? What about the countless other things about Christianity that turn our godless culture’s stomach?
In all honesty, I think many of the things Christians debate these days are devilish distractions that often do little but cause anger, fustration, and turn our attention away from from the author and perfector of our faith and His message for humanity.
In the end, Christians should spend the bulk of their time focusing on Christ as the only answer for racism because we know that protests, politics, discussions about hashtags, and the countless other things we waste time on will never solve a problem that begins in the hearts of men. Racism, as with many of the world’s ills, is the result of sin and until the sin problem is dealt with and people become new creations in Christ (2 Corinthians 5:17) the problem of racism will never be eradicated, even if offensive hashtags are.