We had this come up in a game recently. We were level 10 at the time. The DM ruled that the lava caused ongoing 5 fire damage, and that you couldn't save until you moved out of the lava.
If that seems soft, it's because he put the lava at the bottom of a 50-foot drop. On a rickety bridge. Crawling with creatures that push.
Of course, when the Fighter got hit, failed his save, and fell to the bottom, hovering just a few points above negative bloodied, we thought he was doomed. That's when he got a natural 20 on his death save, stood up, and hauled himself up out of the lava, thereby winning bragging rights for basically the rest of his life.