I would not do that, it's too difficult and unstable to work with. Dry ice is extremely cold, any contact with bare skin, fish, plants ect would cause severe tissue damage or kill them. The other issue is you need CO2 dissolved in the water, and you need to control the dose or amount. Dry ice will loose about 20 percent a day sublimation, keeping it in a confined space ie closed tank creates a chance for asphyxiation.
I agree with the other two. I suppose it could work if you tapped a line into a dewar that holds the dry ice, but that would end up being way more expensive than doing what AnotherHobby suggested, plus it is way more dangerous like jschwabe5 said.
If she is talking about normal ice, then that doesn't even make sense. Normal ice (solid water) carries trace amounts of impurities, but you'd need a room full of ice (containing CO2 impurities) to even get the output of a normal DIY system; that wouldn't last very long anyways.
Above is true when it comes to dry ice, which is solid, frozen CO2. You cannot control the rate at which it melts (actually, sublimates - since it skips straight from solid to gas without ever becoming liquid). Any attempt to confine and save it for later use becomes a pressurized system, with risk of explosion if pressure becomes too great for the container.
But you didn't actually say dry ice. You said ice. Plain old water ice doesn't release any CO2 as it melts. Maybe just a trace, if the tapwater it was frozen from had some CO2 in it, but you'd need a LOT of water.