There hasn't been anything like a confirmation of PS5 remake of the original The Last of Us - its second remaster in less than 10 years, following a 2014 version for the PS4. The project's existence could impact a theoretical Last of Us 3 in several senses.
Details of The Last of Us remake - which isn't officially unannounced - are unclear, but at a minimum it will presumably get true 4K resolution and 3D audio, as well as more detailed characters, objects, and textures. The greater question may be how many gameplay improvements it will incorporate from The Last of Us Part 2, since those could involve rebalancing segments in a way that would demand months of extra work. Bonus content could stretch development even further.
A delay of progress on The Last of Us Part 3 is the greatest potential impact of the TLOU remake, but far from guaranteed. In a recent interview, the reporter that broke the product's existence - Bloomberg's Jason Schreier - said that The Last of Us is "something for people to do for the next year or so" while other games are in pre-production, given that Naughty Dog is a multi-team studio and some workers can temporarily end up off the front lines. It's only if the remake ends up protracted and/or drawing in other teams' resources that a delay of progress would happen. Hypothetically, people could even end up being pulled off The Last of Us remake project to help elsewhere, since no remaster is going to outsell a blockbuster original.
The Last Of Us Part 3 Could Bridge With Part 1
Naughty Dog could also want to incorporate lessons learning from revisiting The Last of Us in a theoretical future Last of Us 3 release. Some changes and additions might be as simple as wanting to copy certain gameplay tropes or set pieces. On a more profound basis, the writers might realize they want to revisit certain themes or plot points, a portion of which are inevitably minimized or ignored in any sequel when it comes time to build a script. Even the best games have to judge what matters, as a script loaded with non-stop fan service might easily be clunky and twice as long.
Creative director Neil Druckmann has dispelled notions that The Last of Us Part 3 is in active development, arguing that it would be "exponentially harder to justify going back to that world" after the previous game, which itself took work to rationalize making. Developers are often hesitant to keep revisiting a series over and over - this can lead to creative boredom or stagnation, resulting in mediocre games or talent leaving for more interesting opportunities. It's entirely possible that The Last of Us remake might be the final "new" entry in the series.