Most Christians believe God and man are different types of being (Mormons are a notable exception). What passages in the Bible support this? Or is it inferred by the doctrine of the Trinity and the doctrine of only one God?
God is certainly higher than man. This passage from Isaiah spells it out quite clearly.
Isaiah 55:8-9 (ESV)
8 For my thoughts are not your thoughts,
neither are your ways my ways, declares the LORD.
9 For as the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.
Of course, being higher doesn't necessarily imply being different. But we have further proof in that God is omnipotent and omniscient.
Genesis 17:1 (ESV)
When Abram was ninety-nine years old the LORD appeared to Abram and said to him, "I am God Almighty; walk before me, and be blameless,
Matthew 6:4 (ESV)
4 so that your giving may be in secret. And your Father who sees in secret will reward you.
Acts 2:23 (ESV)
23 this Jesus, delivered up according to the definite plan and foreknowledge of God, you crucified and killed by the hands of lawless men.
One might still nitpick that omnipotence and omniscience are attributes and don't require being a different type of being. The final point is that God was the first one (as He was before the world and there is only one God). But, one could argue, does the order of existence imply being a different type of being?
I think it's quite obvious that God is a different type of being than man.
But that's arguable—it depends on how the types of beings are defined, i.e. the chosen taxonomy. Similarly, men and natural satellites are quite different, but you could come up with a classification that puts them into the same class.
Along with what dancek posted, I would suggest that Numbers 23:19 speaks pretty clearly that God is not man, but something greater:
19 God is not man, that he should lie, or a son of man, that he should change his mind. Has he said, and will he not do it? Or has he spoken, and will he not fulfill it?
Another verse making this distinction is Hosea 11:9:
I will not execute my burning anger; I will not again destroy Ephraim; for I am God and not a man, the Holy One in your midst, and I will not come in wrath.
God's repeated declarations about himself in Isiah 40-66 are also extremely relevant. That God is the only god is indeed important; the difference between him and all else is categorical. He is not just vastly quantitatively different, but qualitatively different.
All the nations are as nothing before him; they they are accounted by him as less than nothing and emptiness. To whom then will you liken God, or what likeness compare with him? —Isaiah 40:17-18
To whom then will you compare me, that I should be like him? says the Holy One. — Isaiah 40:22
Is there a God besides me? There is no Rock: I know not any. --Isaiah 44:8
Those are just a few examples. I recommend reading chapters 40 through 66 straight through in one sitting to really get a sense of what he is saying.
There are many other doctrines which are only coherent with an understanding that God is more than "a lot different" than man.
- The overall emphasis on the Bible in describing God in the highest possible terms of human language.
- The thrice-holiness of God pronounced by beings far above humans in Isaiah 6. Holiness is a broader concept than righteousness; especially in its threefold emphasis, it indicates a different in being.
- God's identification of himself as "I am" and calling himself by the name YHWH.
- That all sins are so much against sins that they are effectly not even against anyone else. After committing murder and adultery, David says in his repentance, "Against you, you only, have I sinned, and done what is evil in your sight." (Psalm 51:4) Connected to this, God's ability to forgive sins in an ultimate sense.
- The offense of the Pharisees at Jesus' claims to deity.
- Countless other doctrines, such as his omniscience, omnipotence, absolute providence, seeking of his own glory, etc...
Also have a look at God's encounter with Job. To contend that God is only greater and not utterly other is to miss the point of the text.