If you are looking for information on the paintings for the first game, visit this page instead.
Paintings hanging on walls can occasionally be found in dungeons. They have been confirmed to appear only in the first 3 dungeons (basements) and the Challenge Chamber. Two of them are the paintings found in the previous game (Mary and the Explorer with Unknown Motives). There are also new paintings that can be found in this game.
They pose one more threat to the player. If the player stares for too long, the paintings will start burning and if the player continues to stare, they will take damage until the game is eventually lost.
What they have in common is that they all produce sounds when the player is close (for example, Mary is crying). If you have played the game for a while and you hear a constant new sound in a chamber, there's a chance it is coming from a painting hanging on the wall. Look for it and don't stare at it.
The fact that they catch fire is a reference to the way Mary was killed. If we take into account that the half-skeleton painting shows the Explorer with Unknown Motives (the male playable character in the first game), they could be people who were killed, trying to escape these dungeons. They were burnt to death in Room 308, as the cages can be found empty in there if the player gets the Bad Ending. This is only a theory, as not enough clues have been found to prove this.
Examples of paintings can be found in the pictures in this page.