First things first... either take the carts out or flip DIP switch 1 and go into hardware test mode. Press P1 Start until you get to the audio test screen and test the audio outputs. Do you get audio? If so, your audio is working and you need to clean the slots/carts as the Z80 may be crashing because of poor contacts to the sound program ROM.
After that, reset the Backup RAM (part of the hardware tests), reboot the game (or power cycle it), and check again to see if you have audio during the hardware tests now... and if so, if you have audio during the game. Sometimes when the Backup RAM gets corrupt, the audio will disappear.
Next, touch the pins on the amplifier chip when it is on. Do you hear humming sounds when you do that? If so, the amp chip and the caps on the output will be working.
Now it gets more complex...
Do you have a logic probe? Does it have audio capabilities? If it does, you're in luck. Probe the YM3016 DAC chip and you'll hear the beeps change on the inputs to it when the audio is supposed to be playing. If you hear that, then the digital part of the circuitry is working.
If you don't, do you have an oscilloscope? If you do, you can check the inputs to the DAC chip with that.
If not, you're left with randomly swapping out parts until you figure out what was wrong with it.