Perspective shift: the body is not alive.
Once a body is possessed, it dies. It remains a flesh and bone puppet to the possessing entity. The entity can mimic life signs such as breathing and heartbeat but it's not much different to applying electric stimuli to the correct parts of a cadaver - it only appears living.
So, a dead body will naturally start decaying.
There are few other problems the possessed cadaver might face. No wounds would heal, for example. Nicks and cuts and other wounds would keep leaving scars on the body and might force the possession to end early, as it becomes increasingly conspicuous.
Another problem is warm temperature - you would need to keep the body reasonably cold to prolong the time before it becomes unusable. Maintaining the humidity and other environmental factors might mean the possessed body could be kept operational for a very long time.
To maintain the cover, an old person can be a good target. First of all, people don't pay too much attention to the elderly. An old person who lives alone with no other family and not much contact with others for a long time is something people will notice but not immediately think "it's a possessed body". And old people are given a large berth for their eccentricity.
So what if the older gentleman lives alone and doesn't exit his home? And he prefers his home very well r̶e̶f̶r̶i̶g̶e̶r̶a̶t̶e̶drefrigerated air conditioned? It can draw comments, sure, but is within what society is willing to turn a blind eye to. Maybe the older eccentric gentleman is a doctor from a foreign country, too. At this point all the strangeness can be easily accepted.
Until one night the cooling system fails and the dead body falls apart by the time it could be repaired.
Post contains allusions to a short horror story that you can read here. It kind of spoils the twist but...then again, the twist can be seen coming from very far away.