Some ideas:
1. use tape or grips that absorb shock and vibrations (dampens input from the wheels)
2. double tape bars
3. use gel pads under tape
4. raise bars to take more pressure off hands
5. title saddle back a bit to take more pressure off hands
6. try different handlebars
7. try gloves
8. rotate hand positions during ride
9. stretch fingers and hands against handlebar during ride
10. stop and rest at intervals
Sometimes it is not possible to eliminate all pain, so try to minimize pain.