When trading lava with a armorer, does the bucket remain?

Villager Trading

When trading with villagers, console has gotten a huge change to trading. You can see these changes at the Official Minecraft Wiki here. To answer your question, trading a lava bucket to a villager will also take the bucket. Think of it like this. If you wanted a lava bucket and traded an emerald to a player, you'd want the bucket to hold this lava. They can't put lava in their pants without dying.