I don't have the perfect grammarian's answer but I would say it's because the options introduced by "whether" are not statements that are parts of a real, factual scenario. Example: "The governor of a nation should maintain the highest standards of integrity, whether he be young or old." There is not a real person who is either young or old referred to in this sentence. The register is formal. If we wrote "is" instead of "be", that would be much more common but less formal. Most people would consider it correct with "is" and actually be a bit surprised to see "be". I don't think even most English teachers would know this grammar. Compare with: "I don't care whether a governor is young or not, he or she should rule us properly." This sentence is less formal and we much prefer the indicative mood, Using "be" instead would feel wrong, even if a grammarian told us it was "right".

HI, this is because subjunctive mood is used when we are discussing a hypothetical situation. "Whether" can used when we have two or more alternatives, which are hypothetical, therefore you have the subjunctive mood construction.

Is it really a rule? "We're going to the party whether it rains or not." (no subjunctive mode) "Whether he eats dinner or lunch, he brushes his teeth" (no subjunctive mode) In Michael's example: "The governor of a nation should maintain the highest standards of integrity, whether he be young or old." I am not convinced that "whether he be young or old" constitutes a use of subjunctive mode. It's just a list of possibilities introduced by "be", like "be it sun or rain..." "be he young or old..." "be they short or tall..." I think Michael's example also works with "is".