It will reduce mechanical power handling (speaking of just increasing volume) and could possibly reduce output at higher bass frequencies. As the enclosure size becomes infinite, you lower your Qtc down to the Qts of the driver and the resonance point (Fc) down to the free air resonance of the driver (Fs).
Lower Q means a wider range of peak and the lower Fc means more low frequency extension. Also, the bigger the enclosure the less loading on the driver and the more natural it will sound. The bigger you go though, the less power it takes to reach peak mechanical excursion so you can run the risk of damaging the sub if you put the same amount of power to it as you would to it in a much smaller enclosure. This happens because the air in the enclosure acts like the spider and surround do and it helps push and pull the sub back to its rest position.