Whenever governments have a problem with God’s people worshipping the true God as they ought to do, the issue is always competition. Those who have made an idol out of the state do not want the presence of anyone who lives in a way that rebukes that idolatry.
But we must make a distinction between those things an idolatrous state does that are wrong for us to participate in at all, for any reason, and those things which they do because all societies have to do them one way or another, and, because they are idolaters, they are doing them idolatrously. An example of the former would be agreeing to any demands to set your faith aside in certain contexts—while serving on a jury, let us say.
An example of the latter would be voting in elections. For many modern statists, voting is an appeal to the Demos, their god, the people. If we participate in that way, we become partakers of the guilt. But there is nothing sinful in the act of voting, and God routinely uses ordinary means to accomplish His extraordinary purposes.
We have to be careful not to grant too much to those who aspire to Deity. They certainly want you to believe in their omni-competence. Every other movie you see these days features a CIA that is at least as powerful as Zeus, and getting stronger every minute. The great managers and handlers of the populace want you to believe that they completely control the context of all elections, and the limited range of options found in them, but a lot of this is simply hype.
So when you vote, it should be as an act of faith in God. You are not showing faith in them, or in the process, or in democracy. You have faith in Jesus, and whenever the larger society invites us all to register the presence of salt and light, we should do so. Not on their terms, and not the way they intend. They may read it all wrongly, and if they even notice, they probably will read it wrongly. But we should do so anyway. What we do is offered to God as our spiritual worship, and not offered on the altars of democracy.