First off, the winner of the drawing is my friend Angela Danae! Thank you to everyone else who entered, and better luck next time.
Following the NIV footnotes to count quotations of the Old Testament appearing in the New Testament, the most frequently quoted Old Testament verse in the New Testament is.... (drumroll please!):
Quoted 8 times, Leviticus 19:18 – "Love your neighbor as yourself" – found in Mat 5:43; Mat 19:19; Mat 22:39; Mark 12:31; Luke 10:27; Rom 13:9; Gal 5:14; James 2:8
Runners-up (quoted 6 times): Deuteronomy 5:16; Deuteronomy 5:17/Exodus 20:13; Deuteronomy 5:18/Exodus 20:14; Psalm 118:26
What is interesting is that the three responses that I received were all the same, and it was not any of the verses above. The unanimous response was Psalm 110:1! Although my research did not show this to be the Old Testament verse most frequently quoted in the New Testament, it wasn't hard to find evidence that this is commonly thought to be the most frequently quoted verse.
Test Your Bible IQ by Christopher Edwards, page 23 asks, "What Old Testament verse is most frequently quoted in the New Testament?" The answer it gives is the same as the answer I got from all three respondents: Psalm 110:1, which Edwards claims is quoted 18 times.
Bruce Barton in the Life Application New Testament Commentary says on page 480, "In Psalm 110:1, again David was not speaking of himself but of Christ. This verse is the most frequently quoted Old Testament passage in the New Testament (thirteen times)."
Steve Moyise and M. J. J. Menken offer a footnote on page 170 in their book, The Psalms in the New Testament, that documents these eighteen supposed appearances of Psalm 110:1
Mat 22:44; Mat 26:64; Mark 12:36; Mark 14:62; Lk 20:42-43; Lk 22:69; Acts 2:33-34; Acts 5:31; Acts 7:55f; Rom 8:34; 1 Cor 15:25; Eph 1:20; Col 3:1; Heb 1:3,13; Heb 8:1; Heb 10:12-13; Heb 12:2; 1 Pet 3:22
Many of these supposed quotations of Psalm 110:1 contain nothing more than a reference to Jesus being "at the right hand" of the Father or putting his enemies "under his feet". Click a few of the links to read the verses, and you'll see what I mean. While use of these phrases may contain an allusion to Psalm 110:1, I wouldn't consider them quotations. I have no axe to grind, though, so if you have heard that Psalm 110:1 is the "most commonly quoted verse in the New Testament" and you want to stick with that, at least now you know exactly where and how it appears.