Yes, Fiendfyre would’ve killed him and destroyed the soul piece.
If Harry had fallen into the Fiendfyre, the piece of soul in him would die when he did. J.K. Rowling has said in many places that the piece of soul in Harry would have been destroyed if he had died.
7. Why wasn’t the Horcrux inside Harry destroyed when he was bitten by the Basilisk in Harry Potter and the Chamber of Secrets?
A Horcrux can only be destroyed if its container is damaged beyond repair. Harry was healed by Fawkes. Had he died, the Horcrux would indeed have been destroyed.
- F.A.Q. (J.K. Rowling’s website)
Also, Fiendfyre is itself a method of destroying Horcruxes, meaning non-living intentionally made Horcruxes with strong protective enchantments on them, and it’s highly dangerous.
“Fiendfyre – cursed fire – it’s one of the substances that destroy Horcruxes, but I would never, ever have dared use it, it’s so dangerous.”
- Harry Potter and the Deathly Hallows, Chapter 31 (The Battle of Hogwarts)
Harry is never shown as being immune to physical harm - he injured himself playing Quidditch and broke his arm, so it’s just as possible to hurt Harry as it is to hurt any other wizard.
“The Bludger had hit him at last, smashed into his elbow, and Harry felt his arm break. Dimly, dazed by the searing pain in his arm, he slid sideways on his rain-drenched broom, one knee still crooked over it, his right arm dangling useless at his side.”
- Harry Potter and the Chamber of Secrets, Chapter 10 (The Rogue Bludger)
Therefore, it’s almost certain that falling into Fiendfyre and not being rescued in time would kill him, and would also destroy the soul piece residing within him when it does.