Thanks Olivier for finding the schematic!
Here's the power-down circuit.
View attachment 14479
It's controlled by the DPDT swtich at the left. When it's in the position shown, the OFF position, C19 is charged up to the battery voltage. When the switch is turned ON, C19 is connected to R53, and via D14, it is used to provide initial power to IC2A.
IC2A compares the voltage on R53 with a fixed threshold of about 18% of the power supply voltage (set by R52 and R51). When the multimeter is first switched ON, the R53 voltage should be a lot higher. IC2A will drive its output high, turning ON Q2, which turns ON Q3 and feeds the battery voltage to the V+ rail.
After a while, C19 discharges through R53 and its voltage drops below the 18% threshold. When this happens, IC2A's output goes low, Q2 and Q3 turn OFF, and the multimeter powers down.
There are lots of things that could go wrong with this circuit. A good place to start would be to measure (with a different multimeter!) the voltage across C19 when the multimeter is switched OFF (should be equal to the battery voltage), then when the multimeter is first switched ON. It will drop somewhat, but should remain much higher than the 1.6V threshold for several minutes at least.
If that's OK, measure the voltages on C18 (the V+ rail) and IC2 pin 1 (the comparator output) immediately after you have switched the multimeter ON. If they are both more than a few volts, measure the collector and base voltagres on Q2 and Q3.
Post the results here and we can work from there.