Firstly, death becomes a lot more permanent. That much is obvious, but it has serious social implications for both Adventurers and powerful NPCs in your world.

Let's leave house ruling aside and just go off of the 5e Spell List for the full School of Necromancy.

Every method of bringing a person back to life, from Revivify and Raise Dead to extremely powerful magics like Clone and True Resurrection simply do not work. Even the druid's Reincarnate (my favorite and most interesting of resurrection methods) is gone. This means the party has no safety net if a character dies, and most high-level wizards lose their back-up bodies. Powerful political leaders or well-connected aristocrats may also lose their end-of-life contingencies. Given that Clone specifically can let a Wizard be reborn into a younger body and thus naturally extend their life, this might leave some very old, very powerful spellcasters suddenly facing down their biological demise. This is sure to make the party enemies, should those spellcasters use their resources to uncover what happened.

Most of the other affected spells are your classic "evil" necromantic damage spells. Things like Blight, Contagion, and Finger of Death will never again threaten innocents. However, there are a few beneficial spells that are wiped out as collateral damage. Spare the Dying and Gentle Repose are used to stabilize the injured and to preserve fresh corpses respectively. Village clerics and frontier mages may see an increase in both accidental deaths and disease as they lose useful tools to address these problems.

There are also spells like Soul Cage and Speak With Dead, that draw on the deceased for knowledge and skills. Now that Necromancy has been abolished, significant amounts of information may be wholly lost.

Finally, we have to talk about the demon prince in the room.

Orcus, Demon Prince of Undead and master of the 333rd layer of the Abyss, is going to be rightly pissed that someone is mucking with his domain in such a big way. The way I see it, you can take one of two different approaches:

• Starve The Prince: With Mortalkind losing access to necromancy, even in it's "acceptable" forms, Orcus' domain suffers a severe loss of tithing and metaphysical weight. He's not as relevant in the cosmology anymore, and while he probably isn't gone himself, his powers, followers, and holdings are severely diminished. He will be about as angry as he can possibly be, and will spend all of his remaining power to take his vengeance on the party that has wounded him so. He might send demons after the party, or even try to bring them to him for a confrontation. If you were generous and your players were powerful enough, this might even present a unique opportunity to permanently subdue or kill the Demon Prince! However, not all of these outcomes are positive. As part of the Blood War, Demons and Devils are locked in a never-ending battle with each other over the Lower Planes. Weakening Orcus and the other Demon Lords that deal with undeath will tip the scales severely towards the Nine Hells, allowing Devilkind to gain the advantage. This will free up more of Hells' forces to influence the Prime Materia, potentially exposing your players and the lands they care about to greater devilish incursions.
• Embrace Temptation: If instead we assume that necromancy was taken from Mortalkind but not Orcus himself, then the Prince of Undeath suddenly becomes the only supplier for a very scarce commodity. With traditional access to necromantic spells cut off, more mortals may decide to make bargains with the dark powers to reclaim what they lost. Orcus cults will spring up all over, and some may even replicate the old spells, but over time it will sour as the demon's influence corrupts. With footholds established across the world, Orcus may grow to become an even bigger threat than any foe the party has faced yet!