This question is not addressed directly by any portion of Scripture I can think of, nor by our Church's confessions or Book of Church Order. Nonetheless, we as Presbyterians hold to certain Biblical principles regarding worship which can be applied to this matter.
As we say in our Confession of Faith (a summary of what we in the OPC believe the Bible teaches, and which can be found, with Scripture proofs, here), we believe we are to do in worship only what God commands. "But the acceptable way of worshipping the true God is instituted by himself, and so limited by His own revealed will, that He may not be worshipped according to the imaginations and devices of men, or the suggestions of Satan, under any visible representation, or any other way not prescribed in the holy Scripture" (WCF 21.1). In other words, we are to worship God only as he has commanded us to do so in the Bible. (This is sometimes called "the regulative principle of worship," and is also stated in WCF 20.2.) I can think of nothing in Scripture which would command us to give respect to the nation-state within whose boundaries our services are being held during the worship service; the flag being, of course, a representation of the nation which is intended to receive the same reverence due the nation.
Our Confession also says "there are some circumstances concerning the worship of God and government of the church, common to human actions and societies which are to be ordered by the light of nature, and Christian prudence, according to the general rules of the Word, which are always to be observed" (WCF 1.6). While having an American flag in the sanctuary may be common within the borders of the United States, I can think of no "general rule of the Word" which would recommend this practice. Moreover, a very basic biblical principle seems to me to argue against it.
Speaking for himself and Timothy, both preachers of the Word (2 Cor. 1:1), the apostle Paul says in 2 Corinthians 5:20, "[W]e are ambassadors for Christ, God making his appeal through us." While God's ambassador is leading worship, then, the sanctuary is the embassy of our Lord Christ. As my father was a United States Foreign Service officer, I am particularly sensitive to the fact that an embassy is an extension of the nation which the ambassador represents. If there are to be any symbols or decorations in the sanctuary, then, they ought to be reminders of "a better country, that is, a heavenly one" (Heb. 11:16), and not of the particular nation-state in which that embassy of God is located.
Therefore, I can see no reason to put an American flag in a Christian sanctuary, and would object to one being placed in the sanctuary where I regularly lead worship. I hope this has been a helpful answer to your question, and, even more importantly, a reminder that at all times we live under the lordship of Jesus Christ, who alone is our Savior.
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