Updating Operating System to play Diablo 3
Just doing a quick search, we can find the System requirements for Diablo III we see that the minimum macOS requirement is El Capitan (10.11). If you are using Yosemite (10.10) or Mavericks (10.9), this game will not work.



We also see that it's capable of running on Sierra (10.12).  You should be able to upgrade to at least El Cap with no performance issues.

To address your specific questions:


  Does Apple impose an operating system cap because her Mac is too old?


Yes.  In fact, with the release of Sierra, many vintage Macs were left out in the cold.


  Is it advisable to upgrade her operating system to the lastest
  version, or would it be too much of a strain on her system resource?


It's not a requirement to upgrade unless you need to, but given your case that you wish to use a specific application (game), it's necessary.  It's always advisable to upgrade as there are lots of fixes and added features with new releases.