What would happen if you shot a large stack of tannerite with a paintball gun? Would that work?

You'd get paint all over your tannerite.

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It would end up with one or more paint splats.  Not enough kinetic energy in a paintball to do anything else.

Tannerite's ignition is based largely on the velocity of the projectile.  Many slower rounds from firearms won't even set it off - most handgun rounds aren't even fast enough. 

A paintball gun typically fires around 300 feet per second, about 1/6 of what is required to set off Tannerite.  The result would be colorful dots on whatever container you've got your Tannerite in.

I highly highly highly doubt that would do anything but ya never know