I have LibreOffice 5.2.3, 5.2.5, all the way to 6.0.5. Do I need to keep the old versions on my computer?
Who knows your requirements… I, for one, have OOo 1.2 through 3.3, LO 3.3 through 6.1, and AOO 3.4 through 4.1.5… and I need them for testing 
But to get them, I needed to perform a special setup method. Normally, upgrading LO overwrites older version.
You can delete any old version, if you don’t test LibreOffice
Yep, what he said.
Unless you have a specific reason for keeping an older version around - such as development or testing - there’s no reason why you should be keeping old versions of LibreOffice installed; it takes-up valuable storage for no apparent reason and more importantly, it potentially presents a security risk (because you’re running outdated/unmaintained software).