If I kill Immortal Phoenix with Heedless Resurrection, can Immortal Phoenix revive itself?

No, you cannot use Immortal Phoenix to revive itself if it is the unit killed to pay the additional cost for Heedless Resurrection.

According to the RiftJudge FAQ #9523 and Rule 355.15 (targeting rules), this interaction fails for the following reasons:

The Sequence

1. Play the Spell: You announce the play of Heedless Resurrection.
2. Make Choices: You must choose a target unit from your trash to play. At this moment, your Immortal Phoenix is still on the board and is not a valid target in your trash.
3. Lock Targets: Targets are locked in during this stage. You cannot change your selection once the spell has begun the process of being played.
4. Pay Costs: You pay the additional cost by killing your Immortal Phoenix (moving it to the trash).
5. Resolution: Because your target was locked in at Step 2, you cannot retroactively choose the Immortal Phoenix that just arrived in the trash.

While it is true that killing your own unit with a spell can satisfy the condition for Immortal Phoenix's triggered ability ("When you kill a unit with a spell, you may pay [1][C] to play me from your trash"), the targeting rules for Heedless Resurrection prevent you from selecting the Phoenix as the target for the resurrection effect of that same spell.