Does heat damage computers while turned off?

Solution 1:

If the temperature goes above 70-80 Celsius (158 to 176 Fahrenheit) chances are that the capacitors become unstable. A car-trunk in full sunlight can reach that high a temperature.

They can take some heat but not for days on end.

Problem is made worse by old capacitors (old being pre- 2006). They have less tolerance for heat.

Just run the oldest newest of those machines with Memtest86 for 1 hour. If it's unstable it will fail the test.
If one fails I wouldn't bother testing the others and throw the whole batch into recycling.
If it comes through fine you can work your way down the list to older machines or do a binary search.
As soon as you find a bad one you can throw away anything older too.

Solution 2:

You should be okay if you left them in there one or two times. However, prior to booting them up, I would recommend letting them cool down naturally. Otherwise, you may get condensation built up internally which could cause issues. Same way in the reverse (taking a cold laptop into a warm environment).

Think of it like a cold glass of water, taken outside. Most likely, it will sweat. Laptops do not like to sweat.

Solution 3:

If it's enough heat it may melt come capacitors, especially if the hardware (mainly the motherboard) is old, causing shot circuits.

In 32 degrees (Celsius) I don't think it's damaged.