We do, because for us that signifies that they were a member at the time they died. 

Sharon - We keep membership status as "member"(because they were at the time of death) BUT we change another profile field to Deceased Member.

We use "MEMBERSHIP" and "RELATIONSHIP" as 2 other fields that allow us to record with finer distinctions, so that we can run reports with a greater level of detail.

Under MEMBERSHIP, we have options for Baptized Member, Full Member, Former Member (transferred out or withdrawn), Affiliate/Assoc. Member, and Disconnected Member (no longer attending)

Under RELATIONSHIP we use options like New (anyone at the time they are entered to Realm), Visitor, Attending or Engaged, Member of Record... (More)

We leave their Member Status unchanged.  If your question is really about an archival record of who was a member of the church at any given time, that needs to be addressed outside of Realm.  Realm is not designed as an archive support system.