I thought this was a cool question so I thought about it and GitMoe's answer a little bit more:
Dry Ice is frozen CO2. Because it's in its solid state, it's really, really dense, so as GitMoe said, the melting rate at room temperature would produce WAAAAAY more CO2 than needed for a tank (Most people stream 2-4 pea-sized bubbles of CO2 into their tanks per second).
But what if you could control the melt rate? If you had some sort of super freezer you COULD control the melt rate such that it only leaks a LITTLE bit of CO2, but that rate would be really difficult to fine tune. Also, that super freezer is going to be one high end, finicky, EXPENSIVE piece of equipment.
So basically, like GitMoe said, theoretically possible, but completely impractical. We've already solved the problem of delivering CO2 cheaply and efficiently, and it's with CO2 tanks and yeast setups.
Creative thought though!
(BTW, another exotic form of CO2 delivery that comes to mind is pouring soda water into your tank. Yeah it works, but compared to other methods, its a hassle, totally inefficient, much more expensive, not to mention super dangerous to your pH!)