Not easily. As I understand it, the problem is finding bit strings with a sha256 hash whose (256-bit) numerical value is less than a certain number. In addition, part of what goes into the bit string is the bitcoin address of the person who will keep the newly generated coins. I can't imagine a way to exploit that for any useful purpose, sinister or otherwise.
I am a bit confused. Why would this process need to be continually re-done? 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.
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!