Is a IIR digital filter with a0=1 effectively a FIR digital filter?
It's unlikely to be true. IIR filters use feed-back paths and recirculate a fraction of the output (ever-diminishing hopefully). Hence they have the name Infinite Impulse Response meaning that an impulse on the input would cause an output that continues to decay to infinite. 

FIR filter do not have feedback paths and hence the name Finite Impulse Response because the output following an inputted impulse changes and then restores to "normality" after a finite length of time. 

The resulting expression may look like an IIR expression but if it contains y[n-i] then it can't be FIR