How important is it to up-pot in the same type of mix for container figs?  Some people seem to think it's important enough to wash off the root ball bare prior to placing it in a different potting mix ratio.  Is it necessary to do this?

Not necessary at all. 

If tree is in perfect health then not necessary at all. If roots show root damage then rinse and change to faster drainage

The reason some people say that is if there's an interface between 2 very different soil types water can run along the interface instead of penetrating in to the soil.  To avoid that you can blend the soils where they meet or interrupt the smooth interface with things like pine bark chunks.  You can even make numerous small holes in the lower layer and fill them with soil from the upper layer.  I'm sure there are other ways as well.

Well, I have never worried about it and never lost a tree in up-potting.  I go from generic potting mixes to 3:1 all the time and never had trouble.  Seems like mostly a non-issue to me.

If it matters I've have much more success with the health of the tress if I repot prune etc in fall at leaf drop. If I wait till spring I don't get the same production that year.