Sounds like the sustain circuitry on the sound board isn't working properly.

Check that the sustain pot isn't dirty - rotate it full swing either way a few times then put it back where it was.

The 555 timer chip at U7 creates the sustain timing. It's output (pin 3) should pulse to 11 volts for 5 milliseconds when tones are played then go to zero volts. So measure the voltage at pin 3 of U7 to make sure its around zero volts when no sounds should be playing.
This 555 output then goes to the sustain mute circuitry so tones smoothly and quickly decay - the sustain pot is in this circuit if you're looking at the schematics.

Sounds like the problem is U7 and/or the circuitry controlling it.

Not yet, you need to determine if the U7 chip is just doing what its told. So if you measure around 11 volts on its pin 3 output then you'll need to measure its input trigger pin which is pin 2.
A high to low to high pulse on pin 2 will trigger the U7 timer chip output to go high for 5ms. If pin 2 is stuck low, the timer will never time out.

If you measure close to 12 volts on pin 2, then closely inspect resistor R21 and capacitor C16 and their soldering. If ok, then replace U7.

If you measure close to zero volts on pin 2 of U7 then the problem may be around transistor Q2. The transistor is being switched on for some reason and you need to find out why.

BTW, what voltage do you measure on test point TP3 of the sound board?

Your voltages around U7 look normal, 11 volts on pin 3 of U7 would have pointed to a problem around U7 but you measured near zero volts which is ok. Your issue looks further downstream. Do diodes CR1 and CR2 test ok?

Post some clear pictures of this sound board front and back.

Was this sound board ever working in your possession or did you purchase the machine with it malfunctioning?

These diode readings are with your meter set to diode mode and the machine is off right? When you say "0" does that mean no change in meter reading or is the meter telling you it's a short circuit?

Typical forward voltage drop across diodes is around 0.6 measured one way (black meter lead on the banded side of the diodes). Your readings look high especially CR2. If you have spare 1N4148 diodes replace them, if not you can try 1N4004 diodes for test.

Check the voltage on the bottom of resistor R7, it should be near zero volts.
It looks ok but also check that resistor R10 measures 150k. It primarily discharges capacitor C6 to create the sustain decay curve.
As jj44114 mentioned you can replace U8 which mixes the sustain mute with the audio tones. Note, U8 is the same as U1 so if you have IC sockets you could try swapping them. If behavior substantially changes you know that 4049 chip was bad.

I measure around 128k ohms with the R10 in circuit on one of my boards. But a low value for R10 should make the tones shut off sooner rather than sustaining forever. Anyway unsolder one of R10 legs and remeasure it so its not in circuit.

I just measured R10 in circuit on three other boards and they read 130k ohms.. Something in that circuit might be suss.
Anyway change U8 and see how it goes. Note the original U8 chip is a 4049U (U for unbuffered). I can't remember whether its critical for it to be the unbuffered version of the chip or not. i.e. you can buy a 4049B (buffered) or 4049UB (unbuffered). Get the unbuffered to be sure. Don't worry about any extra letters in the part number - they're usually manufacturer specific.

There are two ways the sound board goes silent.
One, certain sounds/patterns finish up by selecting a high frequency 'tone' way beyond what humans can hear. You can see this with an oscilloscope.
Two, the sustain (really it's a decay function, not sustain) gently mutes the current audible tone being played.

So the reason the start game and end game tunes play forever is those tune patterns don't finish with the inaudible high frequency tone. And your decay circuit is malfunctioning meaning the last tone of those patterns is never muted.

What happens to normal sounds and the start/end game tunes if you disconnect one leg of resistor R11 from the circuit?

With R11 disconnected, you've isolated U7 off the sustain circuit so U7 is not your problem.

It sounds like the RT2 sustain pot has lost connection to ground which results in the sustain circuit being unable to mute.

Reconnect resistor R11.
With the game powered on and the last note of the start or end game tune playing, measure the voltage on the lower leg of resistor R12 which is to the left of the RT2 pot. With the RT2 pot adjusted fully clockwise you should measure 12 volts on the lower leg of R12. As you adjust the pot anticlockwise it should end up at zero volts on R12. Is this what you are seeing?

BTW the picture above of your sound board has RT2 set fully clockwise which will play notes on and on. The sweet spot is generally around 11pm to 12pm on the dial.

Earlier in the thread you said the following, so maybe changing U8 fixed the issue but you didn't know it:

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They're totally different boards.

I haven't had to change any LM741 on these boards, but I would first do what barakandl mentions in the below post. Ballys original design of connecting the + input of the LM741 to ground was a bad idea. The input used as a reference (pin 3 in this design) should be connected to half the power-supply voltage which Bally corrected on later boards and what the mod does. This should give a better amplification range.

https://pinside.com/pinball/forum/topic/playboy-as-2518-32-sound-probs#post-3159929

[Edit] I just noticed your sound board already implements the mod. Change the LM741 and see how you go.