Board members with unresolved personality conflict are like trying to think with a migraine. The Scripture reminds us in Matthew 12:25 that a “house divided against itself will not stand.” With unsettled discord, every decision will be tainted, and the church is far too important to expose her future to unhealthy decision makers.
This is where the pastor must take appropriate and biblical leadership. I do not believe that a board can be healthy without a healthy pastor. Before the pastor is a decision-maker, he or she is a shepherd. He or she is one who is committed to the “ministry of reconciliation.” There is no substitute for that.
With grace, I would speak to each one individually and shepherd them toward God’s best, helping them to see something much larger than their offense (or whatever the unresolved case may be) is at stake. If I can increase the primacy of the church and her future, then I can reduce the “pain” enough to bring remedy to each one without penalizing our decisions. If not, then I will ask one or both to step back (which illustrates to them the importance of clear hearts in leaders) but I would stay with them as individuals until the conflict is resolved. Then I would ask them to return.