Will Christ Himself ever become subject to God the Father?
Yes. Scripture teaches that after all things are subdued under Christ, there comes a point when the Son Himself will be subject to the One who put all things under Him, so that God may be all in all. This will occur at a point in time, perhaps at the onset of eternity.
Les explains that this verse is mind-boggling and every word is the Word of God, so it cannot be sloughed off. After all things—principalities, powers, kingdoms, and even death—are subdued under Christ's supreme rule, there is another time word indicating a coming point in time. Whether at the onset of eternity, the Son also Himself will be subject unto Him who put all things under Him, that God may be all in all. Les notes that after He has put everything under Christ's feet, and He has become all in all, yet there is coming a day when even Christ will evidently give everything up to the Godhead. He says he must be careful because highly educated theologians listen, and he does not want hot water unnecessarily, but he will not compromise.
And when all things shall be subdued unto him, then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.