Q: Can't change my my home/work address in contacts.

Might you have restrictions set?

Check Settings > General > Restrictions

I came across this because I ran into an issue where I asked Siri to remind me to get something when I got home and she said she didn't know where I lived.  sigh.  Really Siri?  How is this possible when I know I've put my address in my contact card.  So...on to Contacts I went to update my address that has somehow slipped out.   However, I found though that I couldn't update my main contact card with anything.  That's strange.  What the fix was for my particular issue was that I think I had clicked in Facebook settings at some point and said to Update all Contacts.  When it did that, I bet it replaced my usual contact  card with FaceBook's  info or something along those lines.  This made sense because when I was trying to edit my address/phone on my contact card I did see several FB icons.  To "fix" this I went into Settings > Mail, Contacts, Calendars.  I went to the Contacts section and I clicked on My Info.  I found my old contact card once I searched on my name so I just switched it to that.  I guess the one that FB installs can't be edited?

I had a similar issue in that the Facebook contact was set as My Card.  Every time I clicked on My Card and edited it, it would create another contact.  I then tried to delete the offending card/contact and there was no option to do so.  It was driving me crazy.  Thanks to @PhilWh, I realized that in order to get rid of the undeletable contact, I needed to turn off Facebook having access to my Contact.  As soon I turned it off, the undeletable contact disappeared.  I have set My Card to a card that I am sure that I can edit, and all is good now.  I am going to wait to give FB access to my Contacts again.  I am not sure what the benefits are of having FB access my contacts.  Thanks @PhilWh.

-Stan

No problem, always happy to share what works!  Glad that worked for you.