Replace ALU With Lookup Table?
A single 32-bit x 32-bit to 32-bit lookup table would require an untenable amount of space:

$$ 2^{32} \times 2^{32} \times 32 = 2^{69} \approx \mathrm{5.9\ quintillion\ bits} $$

This is flatly impossible to build. A single microchip can store perhaps 240 bits (~128 GB); you'd need over 500 million of these to store the full lookup table. (Halving the size by exploiting symmetries like a+b=b+a still leaves the size in the implausible range.)

If you want to consider the implications on heat/power and speed, though:


If we generously assume that each of the 500+ million microchips draws 1 mA at 3.3V, you're looking at a total power consumption of roughly 1.8 MW. (Yes, that's megawatts.)
If we assume that each of those chips is 2x2 cm and 0.5 cm thick (including the circuit board), and that they require no other support circuitry, the resulting device will be roughly a 10 meter cube. It takes light about 34 nanoseconds to cross 10 meters; even if we assume that it takes no time for one of these chips to look up a result, this would limit the speed of such a device to roughly 292 MHz.