Christmas carols are the Jurassic Park of Japanese — an isolated environment in which long-extinct vocabulary was revived to roam free. Here's the Japanese equivalent to "Joy to the world," for example:

Morobito kozorite,
Utai matsure!
Hisashiku machinishi,
Shu wa kimaseri,
Shu wa kimaseri,
Shu wa, shu wa kimaseri!

Let everyone gather,
And welcome Him with song!
The long awaited
Lord is come,
Lord is come,
Lord is, Lord is come!

I think it's fair to say that morobito and kozoru are extinct. Shu is at best Christian jargon (I'm not sure if it's still used or not). There appears to be some confusion about whether utaimatsure means "welcome with song" or "humbly sing," (etymologically it all came from the same place) but either way, the imperative matsure is at least rare. Hisashiku is still in use, albeit a bit stiff. The verbs matsu and kuru are alive and kicking, but they are of course conjugated archaically, so that shu wa kimaseri, "The Lord is come," is notoriously opaque to the children who learn it.

There's another way in which this song is similar to a Jurassic Park dinosaur: it is not what it seems. Just as John Hammond's dinosaurs were partly reconstructed with frog DNA, "Morobito kozorite" is in truth not a translation of "Joy to the world" at all. It is a translation of "Hark the glad sound":

Hark, the glad sound! the Savior comes,
The Savior promised long;
Let every heart prepare a throne,
And every voice a song.

See? There's that long awaited, and textual evidence that _utai matsure is indeed supposed to be two separate verbs (sing + enthrone), not a verb and an auxiliary (sing + [+humble]).

Note that there is a Japanese translation of "Joy to the world" that is actually a translation of "Joy to the world". Actually, there are a few; here's one:

Tami mina yorokobe,
Shu wa kimaseri!
Kokoro o sonaete,
Iza mukaeyo,
Iza mukaeyo,
Iza, iza mukaeyo!

Let all people rejoice,
The Lord is come!
Offering your hearts,
Welcome ye Him,
Welcome ye Him,
Welcome, welcome ye Him!

No heaven or nature singing, but clearly much closer to the "Joy to the world" we know and love (seriously, it's the single most beloved carol).

How did this situation arise? Wikipedia has the whole story, but in a nutshell, these translations were made back in the olden days when corporate media hadn't rendered the connections between words and tunes quite so rigid. The first version of "Tami mina yorokobe" was presented set to the tune "Winchester Old", while the words set to "Antioch" (the "Joy to the world" tune) were those of "Morobito kozorite". "Tami mina yorokobe" did have a note, it seems, saying that it could also be sung to "Antioch," but I guess the damage was done.

Incidentally, there's another version where the second line is Mukaematsure! which is, I think, meant as verb + auxiliary (welcome + [+humble]). Perhaps that's muddying the water around utai matsure comes from.