Hmm. It looks like most of the through hole components were hand-soldered, and they didn't clean off the flux at all. I would probably start by cleaning off the board with some rubbing alcohol and a cotton swab, then check for cracked or corroded solder joints.
If you aren't able to find anything obviously broken, then you will want to start tracing back the circuit to figure out what components are driving it. It's hard to make out the traces with all that gunk on the board, but it looks like that LM358 near the buzzer is a likely candidate. the left-side op-amp looks like it may be driving either the buzzer or the turn-off circuitry, probably in combination with those discrete transistors in the top corner. Once you have a basic schematic of the circuit, I would start checking the voltages and maybe some of those resistors.
After that, you would need to get into some real real nitty-gritty circuit debugging, so if the problem does not begin to manifest right away, you may consider getting a new meter before you sink an excessive amount of time into repairing this one.