Anor Londo or Irithyll Dungeon?

[☧ Voievod]

Where to go next after Irithyll of the Boreal Valley ?

[Cake for Mumm-Ra]

Either. They're both dead ends so you'll have to do both eventually. IMO, the dungeon is harder than londo, but the londo boss is much more difficult.

[Azryte]

Either, you have to go to both of them to progress though I usually do the dungeon first since you can get an NPC from there.

[Moon Punisher]

I would go to the dungeon first if you are doing npc quest lines.