Apple Store Diagnostics say my hard drive failed but my computer works fine - Should I replace it?
First and foremost make sure you have a backup regime in place (e.g. using Time Machine).

Also, ensure you're backing up both the SSD and HDD. 

My answers to your questions below assume you do have backups in place!


  Should I nonetheless replace my hard drive?


You're asking for an opinion here, so you risk getting different answers. However, if it was me I'd test both the hard drive and SSD with something better than Disk Utility (but that's easy for me to do as I own a whole range of diagnostic packages). In fact I'd test both of them at least twice, using different software.

Then, assuming all was okay, I'd likely replace the HDD with a fusion/hybrid drive and put the existing HDD in an external enclosure.


  Is there any possible damage the hard drives could have that I can't
  see, despite them working from what I can see?


Yes.