Modern Warfare 3 does require and need a PS Plus subscription to play the game on PS5 and PS4. While you will be able to play the campaign and any local co-op split-screen modes, online multiplayer and online Spec Ops missions will be locked off if you don’t have a PS Plus subscription.
PS Plus can be purchased in monthly or yearly subscriptions and any tier of the service will do. So, you can just go for the Essential level or you can get a bunch of PS Plus Extra and Premium games as well if you go for those higher tiers.
That covers everything you need to know about if Modern Warfare 3 requires PS Plus on PS5 and PS4. For more on the game and how you can jump in with friends, get an overview of Modern Warfare 3 crossplay and some of the best Modern Warfare 3 weapons and guns to use in multiplayer once you jump in.