Minecraft – How to get rid of a Dimensional Door

You can burn them, but it's really dangerous. It can get out of control pretty fast.

UPDATE: Shears should now be used to get rid of leaves. They're 6 times faster than using your bare hands (and 4 times faster than using a sword).

Alternatively, you can use a sword if you don't really care about damaging it:

Leaves can be destroyed faster by using a sword, though swords lose double normal durability when used on leaves. As no other tool yields any increase in removal speed, leaves are best removed with bare hands, or a block or object unaffected by durability, so as not to damage your tools unnecessarily.

Finally, as Jonathan mentionned, the leaves will decay after a certain time if it's not attached to solid blocks (but I don't consider this a 'quick' way to get rid of them):

In Beta, leaves decay according to a simple formula: leaf blocks decay unless they are within 4 blocks of a log block, and connected to the log by other leaf blocks. Thus, harvesting the wood from a tree will cause the leaf blocks to vanish randomly one by one.

Depending on how far along you are, there are a couple of options.

If you're just starting out, you could probably try (carefully) blowing them up with TNT. Just go one at a time, and remember to stay clear of your actual buildings. Wood and leaves have low blast resistance, so it shouldn't be too hard to blow it all up.

If you're a little further along, you could use a couple of mining wells to get rid of the trunk. Alternatively, you could use a quarry, but that's a pretty expensive piece of kit.

On my own world, I recently got rid of one of these by using a mining laser. While I was doing that, I actually did set them on fire, though that might be something specific to the laser.

I've also tried to remove one of these with a Logger, but that didn't seem to work.

Finally, if you want to go the Alchemy direction instead of the technical direction, I'm sure a Destruction Catalyst would do the trick. It's a bit expensive though, and assumes you already have a Philosopher's Stone.