Can I increase the reliability of my hard drive by using less of its' capacity?

Your assumptions are wrong.

The drive has a fixed physical format that is made up of physical sectors, in your case totalling 3TB. By formatting the drive you are effectively clearing the data in those sectors but you are not rearranging them or changing their physical size or layout in any way whatsoever. By formatting at a lower capacity you would simply be wasting the extra space, it would exist but would simply not be being used for anything at all.

Formatting does not change the physical sector size nor space between tracks.

As to unallocated sectors, I believe that drives have a preset number of reserved sectors for reallocation and the drive has no way of telling whether sectors are used by the operating system. SSDs do have a feature called "trim" that can tell the drive the sectors are clear to be wiped but this is something slightly different, spinning platters lack any kind of similar feature. Un-formatted sectors would not be being used for bad sector reallocation.

By formatting a 3TB drive as 1TB you are simply preventing yourself from using 2TB of space and thus wasting 2/3rds of what you spent on the drive. Your 1TB partition would be sitting in front of 2TB of empty space.

The error correcting algorithms used by hard drives, like all other error correcting algorithms, do have a chance of failure. There is an extremely remote but existent risk that the hard drive may have written different data than you sent it. With capacities getting into the trillions of bytes, this chance statistically increases.

But you don't buy yourself extra protection from this by leaving space unused. You do obtain extra protection by storing redundant data, either by using a PAR/PAR2 tool or writing the data multiple times.

However, if you are worried about this, you might as well put the effort into copying the data into another physical drive which has additional benefits like protection of mechanical failure and such.

Not using space would give you extra chances in the event of the hard drive head scraping the surface due to it being struck. But think about it - unless you have a method to space your data evenly sparsely around the disk, assuming you are using this data, the head is likely going to be on or near tracks that contain your data. The days where the hard drive controllers would respond to a "park" command to "get out of the way" of data are long past.

If you format the Hard Drive to its full 3TB capacity and never use more than 1TB, then yes you could potentially increase the lifespan of the Hard Drive. I say could potentially, and not definitely, b/c the Hard Drive could fail mechanically. There is not really anything you could do to prevent that from happening. However, as sectors on the platters start to get corrupted the Partition Table will remap to other sectors on the platters that are not corrupt and not being used. Using a Hard Drive to its full capacity prevents sectors from being remapped, thus allowing for loss of data. Hope this helps! :-)

I note also that you can increase the life and speed a little bit if you partition a large drive then do things based upon a life expectancy of a drive (like rotating car tires) you install operating system on one partition, possibly a very large virtual drive (cache) on one all by itself, and as suggested already the data on the 3rd [assuming you created 3 partitions]. Now if the warranty being 3 years you rotated each partition's use every year so they took turns being written to the most by cache and operating system w/caches and temporary files and such, you accomplish by deleting the virtual memory on 1 partition and installing fresh OS to it, the same as perfect defrag from scratch. You also do so when you copy data from one to the next you can reformat each one clean first, as all files are then copied to brand new drive so-to-speak they are not copied fragmented if you copy them all at once even when they are fragmented on the source partition. Biggest gain is spreading the disk writes across 3 partitions... there are a few significant writes of certain files because an existing filename is always written over the top of itself if it gets larger it adds to the end but all changes start from the beginning. I found weak spots in NDD running years ago trying to read/write over and over before finally writing. I wondered why I had some applications freeze sometimes and realizing NDD if it could write in 6 trys (DOS default attempts) it continued as if everything was ok (not telling me the weak spots) so I used ID app what was being tested clusters... used Norton's Disk Edit to ID what file(s) used that/those clusters... then renamed the physical file FILENAME.BAD. Now missing file(s) had to be reinstalled but guess what...? No more software freezes... technically what I did was manually mark spots on drive bad at dos level, and by renaming them I forced them to be written to a new (strong not weak) drive area. I found files written to multiple times during normal dos opereations were the areas that started failing first most of the time.