Yes, the piece of soul in Harry should still tie the Dark Lord to life.
The point of a Horcrux is to keep a piece of the soul earthbound, so the rest of the soul would be tethered to life even if the body is killed. Usually, removing the piece of soul requires a spell, but there’s no reason that an unintentionally removed piece wouldn’t work the same way - it’s having the soul piece outside the body that tethers the rest of the soul to life.
“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
The additional enchantments on Horcruxes, other than the spell to seal the soul into the object, are to protect it from being destroyed. These spells aren’t what makes the Horcrux tie its creator to life. They don’t have anything to do with the function of the Horcrux, they’re just to strengthen it.
“So does it say how to destroy Horcruxes in that book?’
‘Yes,’ said Hermione, now turning the fragile pages as if examining rotting entrails, ‘because it warns Dark wizards how strong they have to make the enchantments on them.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
Harry being a living person rather than an inanimate object like Horcruxes typically are wouldn’t stop the piece of soul in him from tying the Dark Lord to life, either. Living creatures can be used as Horcruxes, so the piece of soul being in a person wouldn’t stop it from working.
“The snake?’ said Harry, startled. ‘You can use animals as Horcruxes?’
‘Well, it is inadvisable to do so,’ said Dumbledore, ‘because to confide a part of your soul to something that can think and move for itself is obviously a very risky business.”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
In addition, Dumbledore is sure that piece of soul in Harry would ensure the Dark Lord couldn’t die.
“Part of Lord Voldemort lives inside Harry, and it is that which gives him the power of speech with snakes, and a connection with Lord Voldemort’s mind that he has never understood. And while that fragment of soul, unmissed by Voldemort, remains attached to, and protected by Harry, Lord Voldemort cannot die.”
- Harry Potter and the Deathly Hallows, Chapter 33 (The Prince’s Tale)
With the research Dumbledore has done into Horcruxes, it seems very unlikely he’d be wrong about this, and it’s supported by what’s known about Horcruxes and how they work.
While it’s there, ’killing’ the Dark Lord would just disembody him.
Since it would likely keep the Dark Lord tied to life, any attempt to kill the Dark Lord while his soul piece was still in Harry would likely just rip him from his body, like the rebounding Killing Curse did. He’d still be alive, though without a body and in spirit form again - but he wouldn’t truly be dead.
“Aaah … pain beyond pain, my friends; nothing could have prepared me for it. I was ripped from my body, I was less than spirit, less than the meanest ghost … but still, I was alive. What I was, even I do not know … I, who have gone further than anybody along the path that leads to immortality. You know my goal – to conquer death. And now, I was tested, and it appeared that one or more of my experiments had worked … for I had not been killed, though the curse should have done it.”
- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)
Therefore, once the soul piece remained alive in Harry, it’d be impossible to truly kill the Dark Lord.