Okay so I have a theory :
Fyodor's ability is 'CRIME AND PUNISHMENT' and we have seen people die around him in the following two ways:
1. He touches them and they die instantly
2. They harm him in some sort of way (read: lethally) and he switches bodies with them
 A necessary criterion for the above victims is that they were WITHOUT ABILITIES.

As we can see, there are varieties to his ability as well, which leads to the following two conclusions :
1. If someone (without an ability) harms him LETHALLY, they switch bodies with him, inheriting their own inflicted injuries (their CRIME becomes their PUNISHMENT)and he takes up their body, while also transferring his OG body's features.
2. If someone thinks that they have INHERENTLY done something bad, or are doing something wrong OR that they deserve to die, Fyodor inflicts JUDGEMENT on them
 - (this judgement is a mere placeholder, lets call this a name of his ability even, wherein, inflicting JUDGEMENT on a person causes them to die) - causing that person to become free of their CRIME and receiving death as PUNISHMENT. On doing this, Fyodor inherits the remainder of their lifespan. (note: they should be without an ability)

Hence, this can explain his immortality.