I was wondering if there was any canon that would lead one to believe one way or the other. Since the Many-Faced demands a life and one was taken, has the Faceless Man fulfilled his task? Or, since the Many-Faced God is all gods, does R'hollor's allowing him/her to come back just cancel it out?
No, as of now, there is no canon information from the books in this regard because there has never been a case where a man killed by Faceless men was resurrected by priests of R'hllor.
Therefore the following answer will be informed speculation, based on reasoning from canon material:
Who gets resurrected by R'hllor?
R'hllor does not grant his gift of a second chance at life to everyone. Only those people are sent back for whom he still holds some purpose in life. The only known resurrection cases in the books are Lord Beric Dondarrion and Lady Catelyn Stark.
From A Storm of Swords, Chapter 39, Arya's POV:
“Could you bring back a man without a head?” Arya asked. “Just the once, not six times. Could you?”
“I have no magic, child. Only prayers. That first time, his lordship had a hole right through him and blood in his mouth, I knew there was no hope. So when his poor torn chest stopped moving, I gave him the good god’s own kiss to send him on his way. I filled my mouth with fire and breathed the flames inside him, down his throat to lungs and heart and soul. The last kiss it is called, and many a time I saw the old priests bestow it on the Lord’s servants as they died. I had given it a time or two myself, as all priests must. But never before had I felt a dead man shudder as the fire filled him, nor seen his eyes come open. It was not me who raised him, my lady. It was the Lord. R’hllor is not done with him yet. Life is warmth, and warmth is fire, and fire is God’s and God’s alone.”
As you can see, Thoros of Myr had performed the ritual many time and it never resurrected anyone. But when he did it with Beric Dondarrion, he rose up. Because R'hllor was not done with him yet.
It is highly unlikely that someone chosen to be killed by FM would also turn out to be someone who was chosen by R'hllor to do something. It is also unlikely that a man slain by faceless men would have a Red priest around to perform the ritual.
Red God's due & beliefs of Facelessmen
Jaqen said to Arya:
The Red God has his due, sweet girl, and only death may pay for life. This girl took three that were his. This girl must give three in their places. Speak the names, and a man will do the rest.
So in order to repay a debt of life, Faceless Men have to pay back with equal number of lives.
Faceless men give death as a gift. It is not a duty or anything else in literal sense of a duty. Someone goes to their temple and pays ridiculous costs for the gift (Either for himself or someone else) & faceless men handle the rest. If they kill someone and he dies, duty of the Faceless men is done. They have given the gift and that's all what matters.
In AFFC, The Kindly Man tells Arya:
“And many names,” the kindly man had said. “In Qohor he is the Black Goat, in Yi Ti the Lion of Night, in Westeros the Stranger. All men must bow to him in the end, no matter if they worship the Seven or the Lord of Light, the Moon Mother or the Drowned God or the Great Shepherd. All mankind belongs to him... else somewhere in the world would be a folk who lived forever. Do you know of any folk who live forever?”
The correlation between Many Faced God and R'hllor can be stated as ambigious from this source. It is not clear whether they consider R'hllor as another face of MFG or do they consider him a distinct deity. Nevertheless, If they gave the gift to their target and he died, their job is done. They bargained for one life & one life was taken. If R'hllor gives a new life to their target, it should not be any of their concern. The Faceless Men take the count of the slain very seriously (Though exceptions are made if their own life is threatened). From ACOK, When Arya threatens to name Jaqen if he did not help her free the Northmen, He says that it would mean more than three deaths and gods will not be pleased. Nevertheless he agrees after Arya's threat. Afterwards, he said this:
“I take back the name.” Arya chewed her lip. “Do I still have a third death?”
“A girl is greedy.” Jaqen touched one of the dead guards and showed her his bloody fingers. “Here is three and there is four and eight more lie dead below. The debt is paid.”
In conclusion no I don't believe that FM would take a hit on a resurrected man because they have done their job of giving the gift already. If the man is dead, that's good enough for them.
Also, resurrection after death is considered a new life (In show canon), Which is evident because Jon considered himself free of Night's Watch vows after he was resurrected.