I'm not too far in the series but I can't see any use in the prayer yet.
First I thought the prayer either weakened the infernal or boosted the firefighter.
Then I thought the soul needed the prayer to be "freed".
For example, on D.Gray-Man:
A soul can either be freed when killed by exorcist innocence or destroyed when killed using a different method.
Does it make a difference if they pray or not in Fire Force?
Great pressure is pressed on people if they feel they are killing people (or at least I think so). The Fire Force was set up to save people instead of killing, although they may be literally killing all the time.
In order to maintain a sense of justice and mercy, prayers are needed, not only to confort the family and relatives of the dead, but to ease the feeling of guilt of the firefighters.
The people in Fire Force believe that an Infernal is a person who is on the verge of passing on to the next life, and that they are suffering heavily with their body on fire.
By having a Sister say the prayer, they are blessing the soul of the Infernal so that when the Fire Force destroy their physical body they can pass on to the equivalent of Heaven and be at peace. In the first season, the Captain explains how the bright lines on the Fire Force uniforms are both practical (making it easier to see them in smoke) but also symbolic of the Force's role of guiding souls to the next world.
There is at least one point in the second season (and I believe once during the first season as well) where they discuss destroying an Infernal without saying the prayer, and those present are aghast at the idea of doing so since it would condemn the soul of the person - the instance in the second season was a case where they weren't sure that the Sister could arrive in time to say the prayer in a situation where there were many non-Infernal lives at stake so they were weighing up the danger of waiting long enough to do things properly
