Limiting the concept of "lunar new year" to "Chinese New Year" (since there are many lunar and lunisolar calendars, the latter of which the traditional Chinese one is a widely used example), is this photo taken by the Chinese microsatellite Longjiang-2 on February 3, 2019 at 15:20 UTC—slightly less than 25 hours before the Chinese New Year began in Beijing in 2019—the first one ever taken of a virtually fully-lit lunar disk very close to the first day of Chinese New Year?
Since a modern Chinese New Year by definition falls on a new moon (as seen from Earth), the significance of a photo of the whole far side of the moon on a Chinese New Year would be that it's the first time we'd get to "see" a "full moon" on that day.
Note that images like this one from LRO that are mosaics comprised of thousands of photos taken over the course of years—some of which may have been taken on a Chinese New Year but presumably none of which capture anything close to the whole disk of the moon—don't quite fit the bill, beautiful and detailed as they may be:
I haven't found any mentions in the NASA Space Science Data Coordinated Archive's catalog of lunar images of spacecraft that imaged the moon's far side having done so near a Chinese New Year, though that archive isn't necessarily comprehensive.
