How much continuous current is from the battery needed to power my piezo buzzer?
The battery in question would limit current to the buzzer, per the battery's internal resistance. Thus, the buzzer "demanding" more current is not the constraint, it would only get as much current as the battery is able to supply.

Typical piezo behavior with limited current available (and I just tried this to check) is that the volume of the buzzer starts dropping sharply once the available current reduces below around 30-40% of the "rated" value, but it remains audible (faintly) until well under 5% of rated current. 

Also, a piezo buzzer's current demand consists of sharp pulses, much like a capacitor being charged and discharged by a square wave. Thus, the maximum pulse current rating of the battery (10 mA) will come into play, rather than the continuous current rating. 

In short: 

The buzzer will work fine, it just may not be as loud as it can be. Given that even a tiny piezo buzzer (mine has a minuscule 9 mm diameter) makes an awfully loud sound, this may actually be a good thing.