They currently serve no purpose, other than as a trophy or decoration.
Clicking on the egg in survival or creative causes it to teleport to a place nearby (up to seven blocks vertically and fifteen blocks horizontally), creating the same particles as Endermen, Nether Portals, and Ender Chests. It may teleport into the air and subsequently fall to the ground.
Like Sand, Gravel, primed TNT, and Anvils, Dragon Eggs fall when unsupported (due to the fact that they are affected by gravity), and if they land on a non-solid block, (such as a Torch, Slab or Tall Grass) they will drop as an obtainable item. They cannot suffocate the player from falling onto them.
To obtain the dragon egg, the player may push it with a piston or place any non-solid block (like a torch) under the block the egg rests on and then destroy that block so the egg falls onto it. It is possible for the player to hit one while it is still moving to the point it breaks, however this is difficult. You may also use a parried sword to break the egg and get it that way. It is also possible to obtain a dragon egg in Creative by defeating the dragon.
The dragon egg will not burn in the lava upon teleportation or falling.