You're going to tear and rip your hands doing CrossFit, it's inevitable.
Lately my hands have been giving me a shit load of hassle (thanks, Murph); so here's a quick guide to sorting them out, reducing the risk of future tears.
1) Storm into Boots.
2) Buy a callous knife, an extra set of blades and a pumice stone. £10 max.
3) Soak your hands; do the dishes, have a bath, whatever.
4) Dry your hands.
5) Take your radgy new knife and CAREFULLY cut away small layers of the callous. Do not go so deep that you're cutting into raw tissue. It's not a race, take your time. You may end up blunting the first blade quite quickly, so replace as necessary.
5) Dry your hands and get to work with your pumice stone, little circles, until they're lovely and smooth.
6) Admire your belting new mitts.
7) Repeat weekly/fortnightly.
If the process is too painful it might be worth waiting a couple of days to let your hands recover before having another go.