As we all know, difficulty is calculated from the network hashing power.
Since default mining option is to receive rewards, is it possible to forfeit the rewards to prevent an increase in difficulty?
In other words, can you "donate" your rig to BTC for "free"?
You can forfeit the bitcoins while still publishing the block. This will result in fewer bitcoins in existence and would therefore increase the value of the remaining bitcoins that everyone else holds. It would reduce the maximum number of bitcoins that will ever exist. Since the block is still published and difficulty is determined by how fast the previous 2016 blocks were published, forfeiting the bitcoins would not affect the difficulty. Your rig would essentially be working for bitcoin for free. Of course this can only be done if you are solo mining or if you are running the mining pool. If you are mining in a pool that someone else is running, then you don't get to choose the block reward. You could however donate your bitcoins back to the pool if you like. I'm sure the pool operator would appreciate the extra income.
You can forfeit the entire block if you like. This will result in the same thing as simply turning off your rig and not mining, except that you'll still be paying for the electricity. Your rig will be hashing away uselessly accomplishing absolutely nothing useful and costing you money. There would be slightly less hashing power on the network (since your hashing power wouldn't count), and therefore the difficulty would not increase quite as much with the next difficulty adjustment.