It depends on the size and the age of the drive!
A 3.5" is designed to be mounted either on its side or flat with the top of the drive up. While you can run it upside down it's not something I would do as its normal position. In the case of older drives - I wouldn't alter the basic mounting. If you do mount it differently its a good idea to reformat the drive so any slop in the heads will not mis-write or read. Moving the system or banging the drive can crash the drive very easily when its running.
2.5" drives are less sensitive on it's mounting but just like 3.5" drives moving or banging the drive can crash it. most laptops have crash guard protection systems so the head assembly is parked before any damage when the system is banged when running.
In any case make sure the drive mounts are cushioned.