A ruling regarding its interaction with Immune to Events characters is available at the FFG Rules Forum.

Summary: Tears of Lys does not target a character, only places a token on an appropriate character, so the token may be placed on a character that is immune to events. However, the delayed kill effect is still originating from an event card and a character immune to events would be immune to the kill effect when the token is removed at the end of the phase.

Rules FAQ

• “At the end of the phase” delayed effects initiate (and resolve) in an order decided by the first player - regardless of the order the effects that created them were played. In the case of Hear Me Roar and Tears of Lys, the First Player would decide whether that Unique Lannister winds up in the dead pile or the discard pile. ​After this, reactions to the phase ending (or to that poor, poisoned character dying) proceed in the usual order.

• The play restriction of "a character without an intrigue icon" is only checked when Tears is played. Giving the poisoned character an intrigue icon after the fact does NOT counteract Tears of Lys.

• Once you're reacting to "at end of phase" steps, you're no longer "in" the phase. So Tears of Lys kills after/outside of the Challenges phase, but before the Dominance phase. Therefore, effects like Winterfell Crypts ("when a character is killed in the Challenges phase") don't interact with Tears of Lys. For more detail, see the Eyrie's card page.

• Multiple poison tokens must saved from separately, because each poison token comes with a separate "kill" effect. So if a character has one poison token from Tears and one from Tyene, that character will need to be saved twice. For example, you could use a duplicate for one poison token and an Iron Mines for the other.