How can I remove bedrock in survival mode?

Note: This no longer works as of 1.8.2

It's actually possible to break bedrock using dark oak trees. This is only advisable for the nether ceiling as it doesn't involve falling into the void. Once you're on the ceiling (using ender pearls thrown up from as high as you can get in the nether) grow a dark oak tree. Mine out the first level of trunk and plant four more saplings and bonemeal them. If you're lucky, the tree will grow downward and take out some bedrock. You might have to play around with it a bit to get a hole that works for you, but if you stick with it, you can get what you want.

This trick should work fine for bedrock in other places too, but it's pretty hard to be very precise, and it really only works for bedrock below the level of the saplings.

In 1.7.9 you can also place a slab under the bedrock and place a wither skeleton skull on the slab in the half-block space, and the bedrock will be replaced.