As far as I know, there is no verse in the Bible specifically referring to voting. After all, the people in the Roman Empire at the time of Christ and the apostles did not get to vote on the emperor or other rulers. They were simply chosen for them.
However, I believe that there is another practical way to look at participation in government. To vote or participate in government in other ways is an exercise of our citizenship in this world. If we are not to vote, then it is probably because we are to reject full citizenship in our earthly country. This is basically the approach of the Jehovah Witnesses. They refuse to recognize the American flag, vote, or run for office. They think that our citizenship in heaven (Philippians 3:20) rules out full participation in the governments of this world.
Yet, if this were true, then Paul should have set an example for us by refusing to exercise the rights of earthly citizenship. But the truth is, Paul was a Roman citizen and he used his citizenship on several occasions. Consider the following verses:
Acts 16:37 But Paul said unto them, They have beaten us openly uncondemned, being Romans, and have cast us into prison; and now do they thrust us out privily? nay verily; but let them come themselves and fetch us out.
Acts 21:39 But Paul said, I am a man which am a Jew of Tarsus, a city in Cilicia, a citizen of no mean city: and, I beseech thee, suffer me to speak unto the people.
Acts 22:25-29 And as they bound him with thongs, Paul said unto the centurion that stood by, Is it lawful for you to scourge a man that is a Roman, and uncondemned? When the centurion heard that, he went and told the chief captain, saying, Take heed what thou doest: for this man is a Roman. Then the chief captain came, and said unto him, Tell me, art thou a Roman? He said, Yea. And the chief captain answered, With a great sum obtained I this freedom. And Paul said, But I was free born. Then straightway they departed from him which should have examined him: and
the chief captain also was afraid, after he knew that he was a Roman, and because he had bound him.
Acts 25:11 For if I be an offender, or have committed any thing worthy of death, I refuse not to die: but if there be none of these things whereof these accuse me, no man may deliver me unto them. I appeal unto Caesar.
In each of the passages above, Paul used his Roman citizenship for the work of Jesus Christ. We are told to pray for our rulers (1 Timothy 2:1-2). We are commanded to be subject to the higher powers (Romans 13:1-5). And Paul gives us a proper example in using our citizenship in this world for the furtherance of the gospel. I have no doubt that Paul, had he been given the chance, would have voted for the rulers of Rome with a prayerful heart. We ought to do the same.
God has not commanded His people to stick their head in the sand and have no part in the affairs of this world. Though we are looking for a better world, we have every right and responsibility to point this world in a better direction while we are here. We are the salt of the earth (Matthew 5:13). There must be a balance and that balance must always lean toward the spiritual and the heavenly. But God has never kept His people from participating in this life. I do not know if you voted in the last election, but next time, and every time God gives us this privilege, go vote. Do it for the glory of God.