tl;dr> If Jesus isn't separated from the Father on the Cross, then the heresy of patripassianism is true, and more importantly, the notion that God does not change is not.
1. Scripture sets up cases that separate the Father from the Son
Scripturally the idea that Jesus was separated from the Father is typically supported from when Jesus cries out in Matthew 27:46
"Father, Why have you forsaken me?" (a reference to Psalm 22).
To be forsaken implies that indeed, the Father is absent from Jesus in a very real way.
Additionally, 2 Corinthians 5:21 says:
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.
We know that sin cannot abide in the presence of God ([Jude](For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.)), and therefore there must have been a separation.
Of this forsaking, the church fathers wrote:
ORIGEN. But it must be asked, What means this, that Christ is forsaken of God? Some, unable to explain how Christ could be forsaken of God, say that this was spoken out of humility. But you will be able clearly to comprehend His meaning if you make a comparison of the glory which He had with the Father with the shame which He despised when He endured the cross.
HILARY. (de Trin. x. 50 &c.) From these words heretical spirits contend either that God the Word was entirely absorbed into the soul at the time it discharged the function of a soul in quickening the body; or that Christ could not have been born man, because the Divine Word dwelt in Him after the manner of a prophetical spirit. As though Jesus Christ was a man of ordinary soul and body, having His beginning then when He began to be man, and thus now deserted upon the withdrawal of the protection of God’s word cries out, My God, my God, why hast thou forsaken me?
Or at least that the nature of the Word being transmuted into soul, Christ, who had depended in all things upon His Father’s support, now deserted and left to death, mourns over this desertion, and pleads with Him departing. But amidst these impious and feeble opinions, the faith of the Church imbued with Apostolic teaching does not sever Christ that He should be considered as Son of God and not as Son of Man. The complaint of His being deserted is the weakness of the dying man; the promise of Paradise is the kingdom of the living God. You have Him complaining that He is left to death, and thus He is Man; you have Him as He is dying declaring that He reigns in Paradise; and thus He is God. Wonder not then at the humility of these words, when you know the form of a servant, and see the offence of the cross
DAMASCENE. (de Fid. Orth. iii. 27.) Although He died as man, and His holy soul was separated from His unstained body, yet His Godhead remained inseparate from either body or soul. Yet was not the one Person divided into two; for as both body and soul had from the beginning an existence in the Person of the Word, so also had they in death. For neither soul nor body had ever a Person of their own, besides the Person of the Word.
Thomas Aquinas, S., & Newman, J. H. (1841). Catena Aurea: Commentary on the Four Gospels, Collected out of the Works of the Fathers, Volume 1: St. Matthew (958–959). Oxford: John Henry Parker.
2. If Jesus did not separate from the Father, then you are stuck with Modalism or Nestoriansism
2a. Modalism
Theologically, the idea that Jesus was separated from the Father whilst dying on the cross is necessary if one is to avoid the heresy of Patripassianism. Patripassianism states that the Son literally was the Father, so whatever the Son suffered, the Father suffered also. However, the problem with this is that when Jesus dies, it is important that God the Father does not die with Him. If he does, then God has changed state - from live to dead - and is therefore "passable" meaning that he can change. If he can change, the thinking goes, then he cannot be fully 'perfect' because change implies going from a less perfect to more perfect state.
Most directly, Hippolytus wrote against Noetus who held this view.
Patripassianism is sometimes called "Monarchical Modalism," because it is inherently modal in its underpinning. If as Sabelius believed, God is modal, then patripassianism is tautological. If Jesus is God the Father incarnate, then the death of Jesus = the death of the Father. As it is essential that Jesus actually died (and did not just appear to die as the Muslims believe) and if it is equally essential that God the Father not die, since he is perfect, then by definition, Jesus and the Father must have been separated at this point.
Since modalism was rejected as heresy, it was natural to make heretical the idea of patripassiansm.
The book "No one Like Him" explains it well:
Modalism, if adopted, has the further consequence that the Father literally suffered on the cross with Christ. This notion is called “patripassianism,” and initially it may seem innocuous, for all Christians would say that the Father’s heart broke and he empathized and sympathized with Jesus while he was on the cross. But that is not what patripassianism means. It means that the Father, in playing the “Son role” while Christ was on earth, actually suffered and died on the cross. This conclusion seemed inescapably to follow from belief in only one divine nature and in the three “persons” as nothing more than different names that designate different roles or activities played at one time and another. But patripassianism met with strong resistance for a very simple reason. Typically, early Christians believed that God is atemporally eternal. As such, he is absolutely immutable, for change comes with time but is absent from an atemporal being. Moreover, if God is absolutely immutable, he cannot experience changes of any sort, including changes in his emotional and physical state (if he is at all physical). Therefore, patripassianism was clearly objectionable. It destroyed atemporal eternity and divine immutability and impassibility. God could suffer, he could undergo change; and if the divine nature was thoroughly resident in Jesus, then God was subject to time, or so it seemed
Pope Leo the Great, in the Post-Nicene Fathers, directly connects this patripassianism to the essence of the Trinity:
This species of blasphemy they borrowed from Sabellius, whose followers were rightly called Patripassians also: because if the Son is identical with the Father, the Son’s cross is the Father’s passion (patris-passio): and the Father took on Himself all that the Son took in the form of a slave, and in obedience to the Father. Which without doubt is contrary to the catholic faith, which acknowledges the Trinity of the Godhead to be of one essence (ὁμοούσιον) in such a way that it believes the Father, the Son, and the Holy Ghost indivisible without confusion, eternal without time, equal without difference: because it is not the same person but the same essence which fills the Unity in Trinity
2b. Nestorianism
It is possible to make an argument that says only the human nature of Jesus died on the cross - the nature of God did not. This maintains passability, but it presents, in my mind, an even worse problem. It turns God into a monster and doesn't actually involve sacrifice.
Imagine, for a moment that this was "God's plan." First and foremost, it inherently presumes a Nestorian understanding of the hypostatic union - also condemned as heresy. In over-emphasizing Jesus' humanity, it turns him into a dual-natured person. It exalts the created human (who was not, as it says in John 1, with God in the beginning, for it was created)
Leaving that aside, for a moment though, if the Divine nature of God the Son were to have left his human side, we have two unfortunate outcomes.
God himself never actually sacrifices himself in this scenario. He sacrifices a created thing - it's like he tore off a soiled suit, rather than actually descending into Hell.
In this scenario, God creates a human being for the sole purpose of being tortured and abandoned. At the moment of greatest pain and suffering, God abandons him. If only the human is crying out, "Father, why hast thou forsaken me!" then God has turned his back on the best human ever made. Talk about sadistic! Would you follow that vengeful, cruel, and hateful God? I certainly wouldn't!
