All the points above are good. But I'll add one more. The way an automatic cooling system works (both the CPU cooling system, and the case ventilation system) is that it monitors the actual temperature measured at a relevant sensor (inside the CPU chip, or on the mobo). For each of those two systems, the mobo has a pre-set target for what that temp should be, and a set of values for what speed signal to send to the fan(s) for that cooler for whatever temp is measured. (On many systems you have an option to set your own "fan curve" for this if you wish). As your workload changes, resulting in measured temperature changes, the system alters the relevant fan speed signals accordingly. In your hypothetical process, OP, what would happen is that the air entering the case over the ice block would be marginally colder, resulting in better heat removal from the warm items being cooled. Since that would reduce the actual TEMPERATURE at the sensors, this might result in the automatic system reducing the fan speeds since the cooling is better, and then the real operating temperatures would NOT change. All that would change is the fans would operate a little more slowly. It is also possible that the improved cooling would be so small that the fan speeds would not change, and the actual measured temps might drop by a degree or two, but not much really.
Bottom line: you'd gain almost nothing.