If I use the action of ally Faramir to give +1 willpower for my characters committed in quest, and then Faramir is discarded (or killed) during the quest phase by some encounter card effect, does the +1 willpower remains until the end of the phase, or disappear with Faramir ?

The same question applies for ally Arwen and other cards like that.

In this case, you would keep the bonus (Faramir or Arwen). They create a 'lasting effect,' which persists even if the card which created it leaves play / gets its text box blanked. From the rules reference:

Lasting Effects
Some card abilities create effects or conditions that
affect the game state for a specified duration (such as
“until the end of the phase” or “until the end of this
attack”). Such effects are known as lasting effects.
◊ A lasting effect persists beyond the resolution of the
ability that created it, for a duration specified by the
effect. The effect continues to affect the game state
for the specified duration whether or not the card
that created the lasting effect is or remains in play.
◊ If a lasting effect generated by a player card affects
in-play cards (or a specified set of cards), it is only
applied to cards that are in play (or that meet the
specifications of the set) at the time the lasting
effect is established. Cards that enter play (or
change status to meet the criteria of the specified
set) after a lasting effect’s establishment are not
subsequently affected by that lasting effect.
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◊ If a lasting effect generated by an encounter card
or quest card affects in-play cards (or a specified
set of cards), it is applied to all cards that meet its
criteria regardless of their play-state at the time
the lasting effect is established. Cards that enter
play (or change status to meet the criteria of the
specified set) after a lasting effect’s establishment
are subsequently affected by that lasting effect
the same as cards that are in-play at the time of
its establishment. This aids the encounter deck’s
ability to create effective card combinations.
◊ A lasting effect expires as soon as the timing point
specified by its duration is reached. This means
that an “until the end of the round” lasting effect
expires just before an “at the end of the round”
ability or delayed effect may initiate.
◊ A lasting effect that expires at the end of a
specified time period can only be initiated during
that time period.

That is what I thought. Thank you