People that promote this theory use many scriptures as proof texts. They include the Scripture at Calvary where Christ says "why have you forsaken me," the passage in 1st Corinthians where St Paul declares Christ became sin for us, and other passages which state Christ "bore" our sins, and the verse in Isaiah 53 where it speaks of the suffering Servant as being "cut off"
As a note the verse about being forsaken is a quotation of Psalm 22 and Christ is expressing the words of sinners, perhaps the best explanation for this is found by St Augustine of Hippo:
I will deal with the other Scripture verses at a later time."...the first verse of which the Lord Himself uttered on the Cross: "My God, My God, look upon Me; why have You forsaken Me?" For "transferring us in a figure" [1 Corinthians 4:6] to what He was saying, and to His own Body (for we are also "His Body," and He is our "Head"), He uttered from the Cross not His own cry, but ours. For God never "forsook" Him: nor did He Himself ever depart from the Father; but it was in behalf of us that He spoke this: "My God, My God, why have you forsaken Me?" For there follows, "Far from My health are the words of My offences:" and it shows in whose person He said this; for sin could not be found in Him.…"--Exposition on Psalm 44
The main reason the view Christ was separated from the Father is not a viable view if for it's incompatiblity with Trinity theology and the hypostatic union.
For Christ to be separated from the Father would require either Arianism (even temporary) or some form of temporary polytheism. For Christ being God to be separated from God the Father would mean there was some form of fracture in the Trinity. The Trinity is a profession that God is one substance in the Persons that are united. Many Protestants like to assert that Deuteronomy 6:4 (Hear of Israel....the Lord is One) word echad (one) refers to a compound unity. Now if God is a compound unity, then how can God the Son be separated from God the Father and still be One God, that is a compound unity? The answer is plain, it cannot be the case. God cannot be separated from Himself, it would be illogical in anyway. One of the individuals told me it only seems illogical, like when God became a man. However, God becoming a man is not illogical because God the Son did not lose His form as God by assuming a human nature, John 3:13 tells us He was in Heaven while being on Earth. While the theory Christ was separted from the Father is absolutely illogical since it violates the profession of God being One, a 'compound unity.'
Now for the belief that Jesus Christ was only separated from the Father in His human nature alone. This view is less illogical and less blasphemous at the first, but it is still a great error to claim since it would split the Person of Christ into two and undue the incarnation. If He is fully God and man united in one person, Jesus Christ, that is the hypostatic union, then the Atonement would require Him to be separated from Himself, requiring Him possibly to be present solely in His Human nature at the Cross. This person that pushed this theory (the sedevacantist) eventually seemed to admit he did not know much theology (which begs then "how are you a sedevacantist if you don't know much theology" since sedevacantists insist the Church has fallen into heresy, but perhaps he IS a sedevacantist BECAUSE he does not know much theology, but this is the subject of another article.)
In experience I have only met one person (a Protestant) that acknowledged the difficulty in believing Jesus was separated from God the Father at the Cross who accepted it. He said told me in rougly these words "Yes the Trinity was broken up at the Cross!"
To be Continued....