What passages support that God once walked the earth as a man?

As a preface, this doctrine is considered relatively "deep doctrine", which is a term in LDS theology that describes doctrines that are held to be true, but have no impact on our salvation. Other doctrines (not deep) include baptism, faith, repentance, remission of sins, etc as outlined by the 13 articles of faith.

The LDS canon consists of:

• Holy Bible (Old Testament and New Testament)
• Book of Mormon
• Doctrine and Covenants
• Pearl of Great Price- books of Abraham and Moses

The Doctrine and Covenants gives the clearest explanation of some of these matters, and I've also referenced the words of past leaders of the LDS Church, which are held to be authoritative.

Assumptions

To answer this question, a few assumptions must be made:

• God progresses eternally (other answers show scriptures about the eternal nature of God)
• Man is the literal offspring of God
• Man can progress eternally

To prove the first, see this guide on God's eternal nature, and Moses 1:33-35, which is part of the Pearl of Great Price:

33 And worlds without number have I created; and I also created them for mine own purpose; and by the Son I created them, which is mine Only Begotten.

34 And the first man of all men have I called Adam, which is many.

35 But only an account of this earth, and the inhabitants thereof, give I unto you. For behold, there are many worlds that have passed away by the word of my power. And there are many that now stand, and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them.

In this creation is His progression.

To prove the second two, Romans 8:17 is very helpful:

17 And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.

The guide for Man's potential to be like God is also very helpful.

God has a physical body, as tangible as man's

First, we must assert that God has a body. The guide concerning God's body details a lot of evidences, both in the Bible and in other books in the LDS canon.

The most clear manifestation of God's physical body comes in a revelation to Joseph Smith in 1843:

The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us.

Note, there are other verses that hint at God's physical body, but this is the most clear.

As you know, God the Father is rarely mentioned. The classic mention of God being a spirit is John 4:24:

God is a Spirit: and they that worship him must worship him in spirit and in truth.

This verse should be coupled with a revelation to Joseph Smith:

For man is spirit. The elements are eternal, and spirit and element, inseparably connected, receive a fulness of joy;

We can see that spirit is not meant literally. We consist of two parts, body and spirit. The spirit is the literal child of God, and the body is the created vessel that we inherited from our parents. In this sense we are spirits inhabiting a physical body, and in this sense God is technically a spirit. But God is also a soul, consisting of a spirit and a body.

When Christ "gives up the ghost", that means that his spirit is leaving his physical body. Many other churches don't believe this way, but this is standard doctrine, itself defensible, that does not fit the scope of the question.

God was once as man

Now we get to dive into deeper doctrine.

Here are a couple passages taken from another site:

I am going to tell you how God came to be God. We have imagined and supposed that God was God from all eternity. I will refute that idea, and take away the veil, so that you may see. … It is the first principle of the Gospel to know for a certainty the character of God and to know...that he was once a man like us.... (“King Follett Discourse,” Journal of Discourses 6:3-4, also in Teachings of the Prophet Joseph Smith, 345-346, and History of the Church, vol. 6, 305-307, emphasis added)

Mormon prophets have continuously taught the sublime truth that God the Eternal Father was once a mortal man who passed through a school of earth life similar that through which we are now passing (The Gospel Through the Ages, 1945, p 104).

These were given by different prophets (or LDS leaders) att different times (separated by 100 or so years). The words of living prophets is accepted as doctrine and scripture, just like the words in the Bible and the Book of Mormon.

It also makes logical sense. Our eternal nature is to become "like God". How can we be "like God" without having our own worlds and children? That is the aim of every LDS. This does not mean that man can ever become greater than or equal to God. God is always progressing, and we have the ability to eternally progress, so long as we are diligent in this life.

As to God's position as a man, we have no idea, only speculation. The foremost theories are:

• God was as Adam, created by a different God on a different world
• God was as Christ, the savior for his time
• God was as us, a regular person who attained perfection

The second is difficult to defend, because Christ's Atonement is considered Eternal, but this may be the same sense in which God is Eternal. Eternal may not mean "forever"; it may just mean a really long time (like millions of years or something). Eternal is generally accepted as meaning the opposite of temporal, or lasting beyond the grave. This definition is also debated, with some scholars giving figures as to what Eternal represents, and is more fitting for a separate question.