Too much tape, or tape on each shoe, made my knees hurt from trying to keep myself from slipping.
Plenty of people can still move how they want to move in their regular workout shoes. So there is not necessarily a need to go buy new shoes for Zumba unless you decide you want to. Some options are to cut the foot off of an athletic sock and put the tube it creates over your shoes, or do the same thing with a nylon.
Some people like Capezio dance shoes, although they don't have any arch support, so if support is an issue for you, those shoes may not work.
The first shoe change I made was to buy a pair of inexpensive shoes from Payless and I looked for something with a big circle (turn spot) on the ball of the shoe. Those freed up my movement tremendously and I'd probably still be wearing those but I didn't get enough cusion/support in the shoe.
So do some experimenting....try some tape....and let's dance :)