Completed Reminders not syncing to iOS from OS X
Quoting a solution I found in the Apple community web site:


  I was having this problem. I changed the syncing range on the phone (under Settings > Reminders) to sync All Reminders. I then logged into iCloud and opened the Reminders tab. This actually caused the reminders on the phone to match the ones in iCloud.  I then made changes on my phone while that tab was still opened and the changes started to appear in iCould (sometimes takes a few seconds - up to 20 or 30) and synched on my iPad as well. After that, all of my changes started to sync across all devices - iPhone, iPad,Mac and in iCloud, regardless of where I made the initial change.
  
  Hopefully this will work for you too.


Link here.