In Matthew 22:41-46, Jesus raises a question to the Pharisees who were doubting his identity as sent from God. He cites Psalm 110:44, which has the psalmist saying:
The LORD said to my Lord, sit in the place of honor at my right hand until I humble your enemies beneath your feet. [Matthew 22:44, NLT]
Jesus asks them how David could call the Messiah "My Lord" if the Messiah is David's son, and they have no answer.
There are plenty of interpretive issues going on here, but it strikes me that Jesus' argument relies on Davidic authorship of the psalm. Most scholars today deny the authenticity of the psalm headings as later additions. I have a couple contemporary commentaries on the Psalms that do take these headings seriously (I think the arguments in the introduction to Geoffrey Grogan's Two Horizons commentary are excellent, and if I remember correctly Derek Kidner's Tyndale volume takes this approach), but a lot of pretty conservative evangelicals, even inerrantists, don't consider the psalm headings to be a genuine part of the canonical scriptures. The problem with this is that anyone who takes Jesus' teaching as authoritative has strong reasons to accept Davidic authorship of Psalm 110, because Jesus' argument relies on that. So there's a choice between (1) accepting Jesus' teaching as true and accepting this psalm's heading as reliably reporting the truth about David's authorship of the psalm or (2) allowing Jesus to have taught something false if David didn't actually write this psalm.
The only way I can think of to get out of this argument is to consider Psalm 110 to have been written by someone other than David but to express something that, if David had said it, would be true. Then Jesus could give an argument that relies on David having fictionally said something that would be true if he'd said it, and it would therefore have to make sense with David saying it, so his conclusion would follow. But this isn't how those who reject Davidic authorship generally take Psalm 110. They generally take it to refer to God speaking to a Davidic king and a human but not Davidic Israelite (not a king) referring to God speaking to that king as "my Lord". So it seems as if the usual non-Davidic-authorship interpretation still doesn't work if Jesus' teaching is accurate. So even though there does seem to be a third option available, I don't think it reconciles how most who reject Davidic authorship actually take the psalm with how Jesus takes it.