The potential risk of data loss is unlikely, unless the drive was taken offline because it was already failing.
The only reason I can think of the drive losing data by plugging it into a new host and mounting it would be because it was already somewhat corrupted, and when a new fsck runs... it "corrects" the drive into oblivion.
Assuming the old drive was working fine, and the machine it was in previously was shutdown cleanly... I can't imagine you'll have many problems.
the DC is behaving quite rudely and i dont think it will evoke any meaningful response if i asked them whether the hdd was in perfect condition when they removed it from last box and plugged it into new one.
still i have asked them... lets see if they reply...
Plug the drive in... boot up the machine normally.
Assuming bios detects it and there are no critical errors... you are most likely going to be fine.
Once the machine is booted try to manualy mount all the partitions on the drive.
If the previous machine was uncleanly shutdown.. it should warn you that before mounting the partition that it needs to go through a fsck.
Hopefully the prior system was shutdown cleanly... and the partitions will just mount without any command response.
If it does say that they need an fsck...you might want to cancel it or say no... and seek further advice... But, in most cases... it's not going to hurt it to just have the OS do the fsck.