No, 1 Thessalonians 4:16 does not say that the phònè arkhaggelou "voice/sound of an archangel" is produced by Jesus himself. The three prepositional phrases (i.e. en + dative noun) in the verse indicate the attendant circumstance or state of affairs accompanying the action indicated by the main verb. They indicate what sounds will be heard during the Lord's descent from heaven. The sense is clearer if we translate en by "amid" rather than "with":
1 Thessalonians 4:16: "For the Lord himself will descend from heaven amid a cry of command (en keleusmati) — amid the sound of an archangel and amid the trumpet call of God (en phònè arkhaggelou kai en salpiggi theou)".
Another possible way of understanding the passage is to take the attendant circumstance as temporal, i.e. "The Lord himself will descend at a cry of command — at the sound of an archangel and at the trumpet call of God". With none of these are the sounds to be construed as necessarily produced by the Lord. For a biblical text describing a very similar situation with the same preposition, consider the following:
Psalm 47:5 LXX: "God has ascended amid shouts of joy (en alalagmò), the Lord amid the sounding of trumpets (en phònè salpiggos)".
One could similarly translate "God has ascended with joyful shouting, the Lord with the sounding of trumpets," but this does not mean that God is the one who is shouting and blowing on a trumpet. The verses that follow (v. 6-8) show that it is the King's subjects on the earth who are joyfully praising him. The joyful shouting and trumpets are attending circumstances of God's ascent to his heavenly throne, just as the cry of command and the sounds from the archangel and trumpet are attending circumstances of the Lord's descent from heaven in 1 Thessalonians.
There are several other references in the OT to the sounds of cries and trumpets as attendant circumstances of events (though not necessarily employing the same syntax as Psalm 47:5 LXX and 1 Thessalonians 4:16). In Amos 1:14 LXX, the walls of Rabbah burn in fiery blazes "amid cries (meta kraugès)", and in 2:2 LXX, Moab "shall die powerless amid shouting and the cry of a trumpet (meta phònès kai kraugès salpiggos)". In Joshua 6:20 LXX, the walls of Jericho fall flat during the "sounding of trumpets (tèn phònèn tòn slapiggòn)" and a "great shout (alalagmò megalò)". Other attendant sounds are mentioned as well, such as the sound of bells heard when Aaron entered into the sanctuary (Exodus 28:35). Paul also used en prepositional phrases to indicate attendant circumstance elsewhere in his letters. Earlier in the same epistle to the church at Thessalonica, he wrote:
1 Thessalonians 1:6: "You became imitators of us and of the Lord amid much tribulation (en thlipsei pollè), welcoming the message with the joy given by the Holy Spirit".
1 Thessalonians 2:2: "But after we had already suffered and been mistreated in Philippi, as you know, we had the boldness in our God to speak to you the gospel of God amid much opposition (en pollò agòni)".
Here Paul writes that his previous preaching in the city of Thessalonica occurred in a context of persecution that he was suffering, and that the Thessalonican Christians accepted the gospel amid this persecution. The most interesting example however is in Paul's similar use of three en phrases to refer to the moment of resurrection in 1 Corinthians:
1 Corinthians 15:51-52: "Listen, I tell you a mystery: We will not all sleep, but we will all be changed — in a brief moment of time (en atomò), in a twinkling of an eye (en rhipè ophthalmou), at the last trumpet (en tè eskhatè salpiggi). For the trumpet will sound (salpisei), the dead will be raised imperishable, and we will be changed".
Here the attendent circumstance (the blowing of the last trumpet) is more clearly temporal than in 1 Thessalonians 4:16; Paul believes that the bodies of Christians alive at the time of the resurrection will be "changed" into incorruptible bodies at the brief, instantaneous moment when the last trumpet is blown; both events coincide in time.