Do bittorrent programs damage hard disk

The hard disks (even SDD) have mechanical parts or are subject to wearing. The more you use your hard disk from a user perspective, the more you use the hard disk physical parts. So using a BitTorrent client is using your disk. But you don't say you are damaging your car engine because you are driving it, so you don't damage your hard disk either.

The traditional rotational hard disk have different health parameters (that you can monitor via SMART). The main one that you can monitor are:

• Load cycle count: laptop hard disks have a much higher rate than desktop ones due to the more likelihood that they will enter energy save mode.
• Power-on hours: hard disks have a "life expectancy" expressed in power-on hours. Much like human life expectancy, some die young while others die older; it is an average value. So by leaving your hard disk on, you are consuming its life time.
• Reallocated Sector Count: When a sector gets faulty, HDD drives to avoid them after. They "move" the data (when they can still read the data thanks to error correction algorithm) somewhere else. I have a really old drive which has an increasing rate of this parameter, so I know one day it will fails, but it is still working!
• the list is far from complete, but that should give you a good start.

Those two first health parameters can increase by just having your computer turned on with no service running! When a disk is close to breakdown, some other parameters should be monitored. Those are mainly related to fault in reading or writing to the disk. Check the SMART article on Wikipedia.