Are there any Christian groups that follow Matthew 23:9? I was reading in Matthew and saw a fascinating command, directly from Jesus:


  Matthew 23:9 (NWT)
  Moreover, do not call anyone your father on earth, for one is your Father, the heavenly One.


Are there any Christian doctrines or denominations that currently practice this doctrine of not calling anyone on Earth a "father"?
Aside from the Jews for Jesus, I think all Christians take this the way Jesus meant it whether they know it or not. 

I won't copy and paste my answer from What is the Roman Catholic view on Matthew 23:9?, but the gist of it is, the Pharisees took pride in calling Abraham their Father, and the pride is what Jesus is doing away with. 

Not that the Gospels should necessarily be intertwined in such a manner, but if you read:


  John 8:48-58 (NIV)
   48  The Jews answered him, “Aren’t we right in saying that you are a Samaritan and demon-possessed?”
  &nbsp;&nbsp; 49  “I am not possessed by a demon,” said Jesus, “but I honor my Father and you dishonor me. 50  I am not seeking glory for myself; but there is one who seeks it, and he is the judge. 51  Very truly I tell you, whoever obeys my word will never see death.”
   52  At this they exclaimed, “Now we know that you are demon-possessed! Abraham died and so did the prophets, yet you say that whoever obeys your word will never taste death. 53  Are you greater than our father Abraham? He died, and so did the prophets. Who do you think you are?”
   54  Jesus replied, “If I glorify myself, my glory means nothing. My Father, whom you claim as your God, is the one who glorifies me. 55  Though you do not know him, I know him. If I said I did not, I would be a liar like you, but I do know him and obey his word. 56  Your father Abraham rejoiced at the thought of seeing my day; he saw it and was glad.”
   57  “You are not yet fifty years old,” they said to him, “and you have seen Abraham!”
  &nbsp;&nbsp; 58  “Very truly I tell you,” Jesus answered, “before Abraham was born, I am!” 


Jesus calls Abraham 'your father' because the Jews about Him called Abraham 'our father'.  

Throughout the Gospels, Jesus stresses that God is Our Father and that's what's really important.