Does a Christian have a responsibility to participate in government? I have heard it said many times that it is the responsibility of Christian to participate in their government, specifically that every Christian has the responsibility to vote. Is there some biblical justification for this belief? I know that the government of the US is dramatically different than what existed in Biblical times, but is there some precedent set forward by scripture that would validate this?

I am wondering about the US system of government specifically, as it is largely based around citizen participation, and as an American.
No, there is no Biblical imperative for us to participate in government in the way you describe.  The New Testament doesn't say much about our relationship with government, because the emphasis is on our membership in Christ.  We are to be less concerned with things of the world, and more concerned with the things of God.

Here's what the Bible does say about our relationship to government:

Give to Caesar what is Caesar's


  Luke 20:22-25 (ESV)
  22&nbsp; Is it lawful for us to give  tribute to  Caesar, or not?” 23&nbsp; But he perceived their  craftiness, and said to them, 24&nbsp; “Show me  a denarius. Whose likeness and inscription does it have?” They said, “Caesar's.” 25&nbsp; He said to them, “Then  render to Caesar the things that are Caesar's, and to God the things that are God's.”


Jesus is saying here that since we do live under the authority of a human government, we ought to honor that authority.  This is not an imperative to participate in government, but it is an imperative to obey our government (when it does not conflict with our obedience to God, of course).  In this case, Jesus is referring to paying taxes, but I believe that the principle applies to honoring government laws and regulations in general.

Submit to authorities


  1 Peter 2:13-15 (ESV)
  13&nbsp;  Be subject for the Lord's sake to every human institution, whether it be to the emperor as supreme, 14&nbsp; or to governors as sent by him  to punish those who do evil and  to praise those who do good. 15&nbsp; For this is the will of God,  that by doing good you should put to silence the ignorance of foolish people.
  
  Romans 13:1 (ESV)
  13&nbsp;Let every person  be subject to the governing authorities. For  there is no authority except from God, and those that exist have been instituted by God.


Again, we ought to submit to authorities.  And again, there is no indication here that we ought to participate in government - only that we ought to submit to it.

Our true citizenship is in heaven


  Philippians 3:20 (ESV)
  20&nbsp; But  our citizenship is in heaven, and  from it we  await a Savior, the Lord Jesus Christ,


Although we are under the authority of earthly governments, we ought to remember that our true citizenship is in heaven.  Participation in earthly government, while acceptable, should not be our focus.