Your referring to the "not-insignificant number" of 8 million Jehovah's Witnesses?
Jesus answered, My kingdom (kingship, royal power) belongs not to this
world. If My kingdom were of this world, My followers would have been
fighting to keep Me from being handed over to the Jews. But as it is,
My kingdom is not from here (this world); [it has no such origin or
Jehovah's Witnesses support God's Kingdom and consider voting for wordly leaders a compromise of faith.
Another supporting scripture:
Put not your trust in princes, in a son of man, in whom there is no
There are other supporting arguments, but this is the main train of thought. Ultimately JW's understand God's Kingdom to be the solution for all of man's problems, and do not take a share in working out man's issues through wordly kingdoms.
Consider this lofty claim from the UN (JWs believe this to be the foremost manifestation of the "Wild Beast" of modern day).
"They will beat their swords into plowshares and their spears into
pruning hooks. Nation will not take up sword against nation, nor will
they learn war anymore."
Obviously this is not of their origin, but they are claiming to aspire to achieve what God himself has promised through the book of Isaiah, where this is taken from. Isaiah 2:4
One might argue that such a goal is a commendable one, in line with God's "plan". Notice, though, the habit of "anti-God" nations, and how they make such claims:
2 kings 18:31
Hearken not to Hezekiah, for thus says the king of Assyria: Make your
peace with me and come out to me, and eat every man from his own vine
and fig tree and drink every man the waters of his own cistern,
This is a very significant event, because this loud-mouth was representing the king who claimed that no God could stand against him, and he was claiming to be able to give what God himself had promised:
But they shall sit every man under his vine and under his fig tree,
and none shall make them afraid, for the mouth of the Lord of hosts
has spoken it.
Support of the nations in general by allying with their agendas and seeking out solutions through them (ie, voting) is what JW's consider to be a violation of faith in God and his Kingdom.