Why did Jesus submit to the authority of God while Jesus was a man? The Bible tells us that Jesus emptied himself and set aside his heavenly glory and submitted himself to the will of his Father (God).
In order to do the will of his Father (God) who sent him, and to obey Him, Jesus was in a lower position to that of God the Father.
What are the theological implications of this change in position, and was it permanent?
I am looking for a biblical explanation from a Reformed Protestant position.
Whenever the word ‘position’ is used with regard to how the Father, the Son and the Holy Spirit relate to each other, people are inclined to think of “High, higher, highest”. This can be misleading for all three equally have the one nature of Deity. But the three relate to each other in particular ways, as they have functions and roles within the one Being of God. They are utterly united and as one in working out the perfect will of Deity. There can never be three divine wills. There can only be one divine will.
The Reformed Protestant view is that they are co-equal in triunity. However,

Arianism (also, Subordinationism) holds that only the Father is God in
the fullest sense; the Son and the Spirit are ontologically inferior.
According to Arianism, the Son is the first created being.  Ontology
is a branch of philosophy that seeks to answer questions dealing with
reality and existence. (Pilgrim Theology by Michael Horton pp463 &amp; 469
– Zondervan 2011)

The Reformed Protestant view is that the Son is eternal so that there never was a time when he was not; then came a time when he incarnated as the man, Jesus, which did not alter his position as the second Person of the Trinity, nor did it change how he related on earth to the Father in heaven. For example, “Though he were a Son, yet learned he obedience by the things which he suffered” (Hebrews 5:8). Living a human life on a material planet, he could experience first-hand things not experienced in the spirit realm (heaven). His purpose in leaving heaven to be confined for a season to a physical existence was to carry out the Father’s will – a pre-determined will formed in the councils of God before any material creation started – a will that the uncreated Word of God had agreed to before the Word “made everything that was made” prior to becoming flesh (John 1:1-14).
That is why 1 Peter 1:13-20 tells Christians they were not redeemed with corruptible things, but with the pure, sacrificial blood of Christ, “Who verily was foreordained before the foundation of the world, but was manifest in these last times for you.” That is why Christ is spoken of in Revelation 13:8 as “the Lamb slain from the foundation of the world.” Once this was determined in the Godhead, it was as good as done. When the time came to enact the obedience, the suffering, and the sacrifice of this spotless Lamb, Jesus, his obedience to God the Father in all of this was clearly seen – of necessity. It was unthinkable that Jesus, the Son, would diverge from what had been planned prior to creation, yet it remained necessary that the Word made flesh show total submission to the Father. Only a disobedient Son would not submit to his Father. But having been co-equal with the Father prior to his incarnation, he could not be contrary to Deity as God manifest in the flesh, for his nature remained the same, now with human nature added to it.
Yes, “great is the mystery of godliness: God was manifest in the flesh, justified in the Spirit, seen of angels, preached unto the Gentiles, believed on in the world, received up into glory.”  (1 Timothy 3:16) This is said of Jesus, the eternal Son of God, and it is beyond us. These awesome words of scripture tell us what we need to know and to believe, but we cannot grasp it this side of glory. We can only wonder in awe at the willing submission of the Son to the Father, continuing while he walked down here below though then with massively great pressures brought to bear on him by the evil one. What we do know is that had Jesus not submitted to the authority of the Father then, Satan would have triumphed and Jesus would never have defeated him either in the wilderness or at the cross, which would have meant that Jesus could not have been the Son of God. But because he is the eternal Son of God, he perfectly carried out the will of God.