Statues are obstacles which only light can pass through. You can thus see past them, but not move or throw things past them. Breaking a normal statue with a staff of tunneling or a scroll of shattering does not appear to yield anything special.
They're mostly a flavor element, like carpets and bones and such. They sometimes serve as a warning, and sometimes even a trap.
A particular style of Quest Room is a room with lots of statues, and bones covering the ground. This usually indicates that a stronger-than-normal foe (such as a Black Jelly, Goblin Chieftain, Flamedancer, or Vampire) is sleeping in the room. That monster will also be holding the key you need, so the room warnings is more to make sure you don't engage erroneously (and keep your allies out of there lest you end up with 6 Black Jellies on your tail and a dead Ogre).
Some statues are traps that, if you stand near them, they start to crack and then shatter to reveal an actual monster. This is usually an Ogre, Wraith, Naga, or Troll, and are usually found starting at the time you run into Ogres. A potion of telepathy will warn you that a monster is inside of a statue, as you'll be able to sense its presence.
There is another Quest Room which is a room with a key altar that is plain but has many statues. If the key is picked up, all of the statues will start breaking up and revealing monsters, as like the aforementioned trap.
As of version 1.6, there is also now a special monster, the Sentinel, which is a spell-casting statue. It's immobile and tends to be found in clusters. Defeating one will revert it to a normal, obstructing statue.