Is the Son of God eternally subordinate to the Father?

The Bible settles this question definitively. 1 Corinthians 15 teaches on the present and future relationship of the members of the Godhead:

1Cor. 15:25 For He must reign until He has put all His enemies under His feet.
1Cor. 15:26 The last enemy that will be abolished is death.
1Cor. 15:27 For HE HAS PUT ALL THINGS IN SUBJECTION UNDER HIS FEET. But when He says, “All things are put in subjection,” it is evident that He is excepted who put all things in subjection to Him.
1Cor. 15:28 When all things are subjected to Him, then the Son Himself also will be subjected to the One who subjected all things to Him, so that God may be all in all.

In the midst of a discussion of resurrection and the end of the age, Paul tells us that when the 1,000-year Kingdom has come to conclusion, the Lord will finally defeat the last of His enemies, Satan. We read about his defeat in Revelation 20:

Rev. 20:7 When the thousand years are completed, Satan will be released from his prison,
Rev. 20:8 and will come out to deceive the nations which are in the four corners of the earth, Gog and Magog, to gather them together for the war; the number of them is like the sand of the seashore.
Rev. 20:9 And they came up on the broad plain of the earth and surrounded the camp of the saints and the beloved city, and fire came down from heaven and devoured them.
Rev. 20:10 And the devil who deceived them was thrown into the lake of fire and brimstone, where the beast and the false prophet are also; and they will be tormented day and night forever and ever.

Back in 1 Cor 15 Paul says that all things have been put in subjection to Christ by the Father. In 1 Corinthians 15:27 Paul quotes from Psalm 8, which relies extensively on pronouns, so it’s helpful to replace the pronouns with the names of the Godhead:

1Cor. 15:27 For [the Father] HAS PUT ALL THINGS IN SUBJECTION UNDER [Christ’s] FEET. But when [the Spirit] says, “All things are put in subjection,” it is evident that [the Father] is excepted who put all things in subjection to [Christ].

Paul says that when Psalm 8 testifies that the Father places all things in subjection under Christ’s feet, it’s clear that the Father Himself is never subjected to Christ. Paul makes this statement to ensure we understand that Christ always remains subjected to the authority of the Father within the Godhead. So the Bible testifies clearly that during this age and during the Kingdom age that follows, the Son remains subjected to the Father. 

However, Paul goes on to say in v.28 that once Christ has fulfilled His mission to rule over all things, then Christ returns all authority to the Father, and then Paul says the Son Himself will be subjected to the Father. Again, it’s helpful to replace the pronouns in Paul’s writing with proper nouns:

1Cor. 15:28 When all things are subjected to [Christ], then the Son Himself also will be subjected to the [Father] who subjected all things to [Christ], so that God may be all in all.

While we don’t understand this statement fully (i.e., what does it mean that God is “all in all”?), nevertheless we can clearly see that Christ remains subjected to the Father even after the Kingdom is over. So the submission of the Son to the Father is clearly spelled out in Scripture.

Regarding complimentarianism, the Bible refers to Christ’s submission to the Father as a model for our submission to Christ, and a wife’s submission to her husband:

1Cor. 11:2 Now I praise you because you remember me in everything and hold firmly to the traditions, just as I delivered them to you.
1Cor. 11:3 But I want you to understand that Christ is the head of every man, and the man is the head of a woman, and God [the Father] is the head of Christ.

So once again, the teaching of the Bible is consist on these points, and we believe unambiguous: within the Trinity, Christ is submitted to the Father. For a longer discussion on the mystery of the Trinity, we recommend you read the following:

Explain the Trinity
What evidence is there for the Trinity?