Does Janna heal the damage from the Elder Dragon, making Elder Dragon not kill them?

To answer your question: No, healing your unit with Janna will not save it from the damage dealt by Elder Dragon’s play effect.

Reasoning

This interaction relies on how Elder Dragon's passive ability modifies the rules for lethal damage.

1. The Triggered Ability: When you play Elder Dragon, its "When you play me" ability is placed on the chain, targeting your unit.
2. The Passive Ability: Elder Dragon’s passive ability states that "Any amount of your damage is enough to kill enemy units." This modifies the definition of lethal damage for any unit that has damage marked on it by the Elder Dragon's controller.
3. The Resolution:
   • You play Janna in response to Elder Dragon’s triggered ability.
   • Janna's ability resolves first: She heals your unit, removing any pre-existing damage.
   • The Elder Dragon's triggered ability then resolves: It deals 1 damage to your unit.
   • Because the Elder Dragon’s controller is the source of this damage, the passive ability recognizes that the unit now has damage marked on it by that player.
   • Per the modified definition of lethal damage, that 1 damage is sufficient to kill the unit.

Healing the unit before the damage is dealt does not protect it because the damage event itself creates the condition—marked damage from the Elder Dragon's controller—that triggers the modified, instant-kill lethal threshold.

Sequence of Events

1. Play Phase: Opponent plays Elder Dragon. Its triggered ability is placed on the chain, targeting your unit.
2. Reaction Window: You play Janna in response to the Elder Dragon trigger.
3. Resolution:
   • Janna resolves: Your unit is healed.
   • Elder Dragon trigger resolves: Your unit takes 1 damage.
4. State Check: Because the unit now has damage marked on it by the Elder Dragon's controller, the passive ability applies. Since the unit has non-zero damage that is considered lethal, it is killed.

Note: This interpretation is based on the specific interaction logic detailed in RiftJudge FAQ #10289 and #9964.