Up until recently, we have only seen avatars lavabend. So we weren’t sure as to which element took on lava. Fire or earth.
Now we have confirmed that lava is indeed an earthbending subcategory, but some still question if firebenders can lavabend, too.
My answer right now is no. I might be wrong, but let’s look at this scientifically.
Lava is essentially really hot earth. Earthbenders work lava the same way as water/ice benders. They increase the friction of the atoms within the elements to generate heat, or to take it away. When you heat up ice, it turns into water. Same with earth.
The reason why lava is so hot is not because it has fire, but because the melting points of all stones and crystals are much higher than that of ice, and therefore require more friction and heat in order to melt. Since more heat must be generated, then lava is very hot.
"But if firebenders can’t bend lava, then why could both Toph and Katara bend mud?"
Like I said. Lava isn’t fire, and it doesn’t have fire in it. The reason why Toph can bend mud is because she bends the earth in the mud. Katara bends the water in the mud. You can separate the earth and the water by boiling the mud. You can’t do that with lava. You can’t separate the fire from the earth, because lava isn’t fire and it doesn’t have fire in it.