Once you have generated all the hashes of numbers less than 256 bits long, then you instantly (up to the speed of accessing the database entry) know a short preimage for every hash value.
Generating all those hashes and storing them? Good luck. 2^256 is about 10^77. For comparison there are on the order of 10^50 atoms on Earth. There's no way you are making a look-up table for every number less than 256 bits.
Rainbow tables allow you to choose a tradeoff between storage and speed. So if you wanted to do that (which is of course unrelated to Bitcoin), the actual constraint is: (table size) * (number of hash calls to invert one hash value) = 2^256.
Now you can donate to Singularity Institute using Bitcoin.
Currently Bitcoin mining appears to be profitable as only bubble economics can be. Already some Less Wrong users have purchased GPUs and started mining Bitcoin. Please consider sending some Bitcoins to SI at address 1HUrNJfVFwQkbuMXwiPxSQcpyr3ktn1wc9
2014 Edit: Please donate Bitcoin by using the Bitpay link on MIRI's donate page. Thanks!