This post is an extension of Songs from the Psalms. So here is a list of hymns in the New Testaments and songs I know for them. (I forget where I got the list.) The '+' sign denotes songs I have found to be helpful in practicing a Psalm of the Day.
Lk 1.46-55 (Magnificat)
+ My soul proclaims your greatness, Oh God, Haugen
Magnificat anima mea Dominum
Lk 1.68-79 (Benedictus)
Lk 2.29-32 (Nunc dimittis)
Nunc dimittis (Wartburg Choir)
Now that I've held Him in My Arms, Michael Card
Lord, now you let your servant go in peace (Lutheran liturgy)
Col 1.15-20
+ Phlp 2.6-11
+ Carmen Christi, Michael Card
The Exaltation of Christ (Wartburg Choir)
1 Tim 3.16
Eph 5.15 Awake, O sleeper
+ Rev 4.8 Holy, holy, holy, Michael Card
+ We Fall Down, Chris Tomlin
+ Rev 5.9-10 You Are Worthy, Michael Card
+ Rev 12-14 Worthy Is the Lamb, Handel
+ Rev 11.15; 19.6, 16 Hallelujah Chorus, Handel
Rev 11.17-18
+ Rev 15.3-4 Song of Moses, Michael Card
Rev 19.1-8 Hallelujah, Michael Card