Does heat damage computers while turned off?

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.

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.

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.

no, a mid heat summer cannot damage your components. It will work 100% Many components can withstand heat upto 200deg F such as a cpu and a graphics card.