Yes, when mechanical hard drives are manufactured they are assembled in a clean room, but they are not airtight. You can remove them from their enclosure and run them but you will expose the platter(s) to dust particles/humidity and this will eventually lead to failure.
That said, if you are doing this for data recovery then it really is not going to matter much. You will no longer have a clean read/write surface, but the whole point of this seems to be that you want to get the data off the drive before it fails completely (if I am not mistaken).
After clarifying exactly what you are referring to by bare drive, I should point out that cooling probably will not be an issue. For a long time I had a WD VelociRaptor 2.5" drive sitting outside the chassis of one of my computers connected via external SATA. This drive runs at a demanding 10K RPM but only requires passive cooling. The drive itself is 2.5", but it has a heatsink built-in that makes it occupy a 3.5" form factor:
Your drive should not require anywhere near the same level of sophistication in terms of cooling as the WD VelociRaptor. But it would be a good idea to put it somewhere with good ventilation since you are going to run the disk continuously to get the data off of it.
If you ever take apart a TiVo and look where the disk drives are mounted with respect to cooling fans, you will notice there is no special consideration taken to cool the drive (short of not putting it directly over top the CPU or PSU). The only fan in a TiVo is always located next to the PSU, and blows air out.