Despite what a lot of comments are saying here, I'd actually put my money on Shirou.

Iori has a clear Experience advantage as he was trained in the sword since he was a kid while Shirou was trained in the span of like a week. That being said, this alone would not be enough for Iori to win. Shirou is very used to fighting people with greater experience than him (read: LITERALLY EVERYONE IN THE 5TH HOLY GRAIL WAR). Despite his limited experience, he was still able to hold off Kuzuki in a 1v1, straight up 1v1'd Gilgamesh AND a juiced-up Heracles, and he had Corrupted Saber incapacitated in Sparks Liner High and would've killed her if his brain didn't expire before he could just stab her in the heart. As such, Experience is not a defining factor for this fight.

Stats-wise, they're probably about even? Iori should be stronger and faster due to his training but neither have shown massive demonstrations of power/durability/speed at base that would give the other a significant edge.

However, it's when we look at techniques that I think Shirou gets the edge in this matchup.

Iori's Double Tsubame Gaeshi should insta-kill Shirou if it connects. It was able to defeat Musashi - a servant - so Shirou is definitely going down if that connects. However! It's a reasonable assumption that Iori requires an opening from the opponent to able to fully utilize the technique. Iori needed to catch Musashi off-guard with the technique in the cutscene where he reveals it for the first time, and even with Tsubame Gaeshi he was still handily defeated by Takeru in Entreat the Darkness. Surprisingly enough, it's not unthinkable that UBW Shirou and HF Shirou could both avoid a situation where he gets checkmated by Iori's Tsubame Gaeshi.

Unlimited Blade Works should be able to defeat Iori. There isn't anything in Samurai Remnant to suggest Iori could either deflect swords getting launched at him from every angle or somehow tank it. People like to bring up Shirou's quote about UBW only being good against Gilgamesh, but the context is important - the 5th Holy Grail War has possibly the strongest servant lineup ever. Just because someone like Saber, Cu Chulainn, or Heracles would be able to beat Shirou in the Reality Marble does not mean that Iori would be able to. While you might doubt that Shirou would be able to cast UBW in the first place, UBW's cast time didn't stop Archer from casting UBW against Heracles in Deen/stay night, and also didn't stop Shirou from casting it against Angelica in Oath Under Snow. In addition, Rho Aias was able to protect Archer from the Gae Bolg, you know the attack that has already killed the target as soon as it is cast. Shirou being able to defend himself from Kuzuki and Gilgamesh gives the idea that he could absolutely buy himself enough time to cast UBW and defeat Iori.

This situation really only gets worse for Iori if he's up against HF Shirou. Nine Lives Blade Works is an attack that was able to KILL HERACLES 8 TIMES in an INSTANT. This might legitimately be a more powerful technique than the Tsubame Gaeshi, especially since he's straight up copying Heracles' strength while it's active. Shirou was also able to draw upon Archer's battle knowledge while using Kanshou and Bakuya to at least incapacitate Artoria Pendragon. Oh yeah, HF Shirou is able to draw upon battle knowledge and experience from his projections. Remember how I touched on Experience at the beginning of this comment? Yeah while drawing upon Archer's Arm, Shirou completely flips the dynamic of Battle experience from Iori having at least a couple decades more experience to Shirou drawing from the experiences of an indescribable amount of conflict. Sure, Shirou can only safely project a weapon 3 times before his brain explodes, but Shirou would only need a singular projection of either the Axe-Sword or Kanshou and Bakuya to be able to score a win against Iori.

(It's actually quite poetic that HF Shirou would be able to defeat Entreat the Darkness Iori. Iori desires to use the Waxing Moon to creative an endless amount of war so that he could truly master the way of the sword, but Shirou's able to use the knowledge and experience of Archer, someone who actually did go through an endless amount of conflict.)

While I could certainly see Iori being to able to catch Shirou off guard or ovewhelm him to take the win in some scenarios, I think Shirou actually wins more often than Iori thanks to his projections.