I am leaving Taiwan, permanently, on June 30 with an ARC that expires July 31. Will I be required to show something as proof of paying taxes for 2007 in order to depart…or show anything at all regarding taxes?
You don’t need anything special.
nobody leaves taiwan “permanently” do they?? 
The computors are definately linked somehow (they can print out your entry/exit dates at the tax office), however I’ve never heard of anyone actually being stopped.
If you have filed your 2007 taxes you could always just take the reciept slip with you just in case (the bottom of the big tax form you fill in).
I have. The tax office can place someone on a list if they have delinquent tax bills and the immigration officers will not let someone on the list leave, and I know someone who had problems leaving because of this. However it is unlikely you will end up on the list without knowing about it, and it takes a quite over due tax bill before you’d be listed.