You don't have a pullup between the two inverter sections.
But your heating problem is that you don't have any current limit resistors on either the input or output of the demo optos on the right end of the first picture. That's why the inverter is getting hot, no current limiting.
To be clear, the DEK opto input has built in current limiting. Your demo optos have to have external current limit because they don't incorporate any internal limiting.
Replace two green wires feeding the LEDs of the demo optos with 150 ohm resistors to limit the current. Add 150 ohms to the LEDs in series on the opto outputs too for the same reason. You likely have smoked the inverter and the demo optos.
I am assuming that the demo opto LEDs are powered by +5V. 24V will smoke the LS04.
But your heating problem is that you don't have any current limit resistors on either the input or output of the demo optos on the right end of the first picture. That's why the inverter is getting hot, no current limiting.
To be clear, the DEK opto input has built in current limiting. Your demo optos have to have external current limit because they don't incorporate any internal limiting.
Replace two green wires feeding the LEDs of the demo optos with 150 ohm resistors to limit the current. Add 150 ohms to the LEDs in series on the opto outputs too for the same reason. You likely have smoked the inverter and the demo optos.
I am assuming that the demo opto LEDs are powered by +5V. 24V will smoke the LS04.