Problems with Atari Punk Console circuit

Using a TLC555 should not be a problem. The behaviour you observe is a failing of the APC circuit itself.

When the rightmost potentiometer is at 0 resistance, the 555 heats up rapidly because, internally, pin 7 shorts to ground during parts of the oscillation, to discharge the 0.1uF capacitor. With the potentiometer at 0 ohms, this results in a direct short across the power rails and a very large current flowing through pin 7 (as well as a direct short discharging the capacitor). The absolute maximum current through pin 7 for the TLC555 is 150mA. To ensure this isn't exceeded, you could try connecting 220 ohm resistors (I'm assuming a 9V supply) in series with both potentiometers, and also between pins 6 and 7 of the rightmost 555.

Most of the audio range of the circuit is contained within the lowest 20% or so of the resistance of the right-most potentiometer. Try reducing the value of this potentiometer, or connecting a 10k 'fine tuning' potentiometer in series with it, if you want more precise control.