Hi good people.

I'm playing Targ. I incinerate an opponent's character, reducing its strength to zero. It is killed. Now I wish to use my Rhaenys's Hill. My opponent claims the character that I just killed still has zero strength. This strikes me as absurd, but I can't find anything to support my position.

Is it still zero? Does "until end of phase" persist unto death? Why/why not?

If the character has left play (which it has to have done at the point you're able to play the Hill), the burn effect is gone.

I believe that to be true. But I couldn't find anything in the rules indicating that it was the case.

If you take 3.16 from the FAQ and the specific explanations under the "What is moribund" entry, it states that when a card leaves play, it no longer has an affect on the game (unless it's effect is explicitly in effect while it's out of play). Once that card leaves play there really is no way to know that the exact same card is returning to play. Any abilities on the card are no longer in effect and anything that was directly affecting that card before it left play would no longer apply. Even "until end of phase" effects that directly affected that card could no longer apply because the card that it affected is no longer in play. The card that returned to play is, for all intents and purposes, a different card.

The other place to look is the FAQ entry 3.14 about out-of-play states where it says "In general, card effects on non-event cards in a player's hand, deck, shadows area, discard pile, and dead pile are not considered to be actionable unless the card specifically states that it can be triggered while in its out-of-play state." This not only means that you cannot use the effects on out-of-play (non-event) cards, but also that out-of-play cards cannot be targeted or affected by cards unless they specifically say so. The entry also gives the example of the CS-Khal Drogo coming into play with the condition "return to hand at the end of the phase" - and that condition being ignored, wiped out, if Khal Drogo leaves play before the end of the phase (so you cannot return him to hand from your dead pile, for example).

When a card leaves play, all effects and conditions on it are ended - because cards that are not in play cannot affect or be affected by cards, effects, and abilities unless something specifically says so. As doulos2k says, this means that if a card "returns" to play, it is treated as a brand new copy of the card, having no "memory" that it was in play earlier.