No, He is not, though Covenant theologians try to say that He is. But they do not read carefully. Psalm 110:1-2 make it clear that Christ is presently seated on His Father's throne in glory. The LORD said to David's Lord "Sit at My right hand, until I make Your enemies a footstool for Your feet. The Lord will stretch forth Your strong scepter from Zion, saying, 'Rule in the midst of Your enemies.'"
Both words for Lord are used of God. The first, LORD, is Yahweh, and the second, Lord, is Adonai. Here in this passage Adonai is used to describe the Lord Jesus, Israel's Messiah. To be seated at the right hand of the king was a mark of distinction (1 Kings 2:19). Christ quotes Psalm 110:1-2 to the Pharisees (Matt. 22:41-46). It confounded the Jews because they knew that Adonai referred to David's Son, the Messiah. "And no one was able to answer [Christ] a word, nor did anyone dare from that day on to ask Him another question" (v. 46).
Jesus is now seated on His Father's throne in heaven, as Psalm 110 says, and, as Revelation 3:21 indicates. It reads: "He who overcomes, I will grant (give) to him to sit down with Me on My throne, s I also overcame and sat down with My Father on His throne." Christ's throne is the Davidic throne that will be established in Jerusalem during the Millennium. "I will grant" is a Future Tense. We are not seated now on Christ's throne, nor is He sitting on it until the Millennium begins after the seven year tribulation.
In Psalm 110:1 the Hebrew word "until" (Ay'th) is virtually the exclusive meaning. And, it means just what it says. Christ is seated on His Father's throne UNTIL the Lord does something else, and that is, until the Messiah begins His reign in Zion, meaning Jerusalem. Psalm 110:1-2 are the most quoted verses in the NT from the OT.
Paul reminds us that after Christ's resurrection He was "seated at [God's] right hand in the heavenly places" (Eph. 1:20). He has been given all rule and authority and power and dominion presently (v. 21). That is, Christ now shares universal authority over all of creation with His heavenly Father, because He is seated by His Father on His Father's throne. However, in terms of literal measurable history, He is not yet seated on His father David's throne, on earth, in Jerusalem. That is coming next. In Providence, Christ is co-reigning over the universe but in terms of His messianic authority over Israel, that is not happening yet because the Jews are not back in the land in faith, as they will be when the Millennium begins.
One cannot make a leap, as the Covenant people do, to say that we are reigning over the universe with Christ. Christ is "head over all things to the church" (v. 22). To make this verse make sense you need to look at it this way: "The Father gave Him … to the church." Or, "He is head over all things that relate now to the church."
John Piper mis-states when he writes:
"Therefore, Ephesians 1:23 would mean: Jesus fills the universe with His own glorious rule through us. We are the fulness of His rule. We rule on His behalf. In that sense, we sit with Him on His throne."
Paul, nor any of the other apostles say this!
Christ's throne has to do with the Jewish messianic kingdom rule. --Dr. Mal Couch