Can a too powerful battery damage a circuit?
The current rating of a power source (your battery) is the maximum current the supply can provide. 
The current rating of a power load (your IOIO board) is the amount of current it can demand, for proper operation. 

Thus, if the IOIO board just requires 10 mA, and operates at a specified voltage range (5-15 Volts as mentioned), then the power supply can not force more current through the board than demanded - it will merely supply as much current as is demanded, not the maximum that it can supply.

So if the voltage values in the question are correct, then the IOIO board should be perfectly safe to operate with the battery.