I bought a 24v electric heater that's capable of drawing up to 10a. It's wire in a flat silicone pad. I intend to make a heated mirror with it. I have the mirror, the heater and the adhesive, but don't know how to supply power to it. An inexpensive 24vac 40va adapter heated it to about the right temp and kept it there for 3 or 4 minutes before burning out.

Someone suggested I buy a 110vac to 24v transformer (I assume AC, since I can't find DC), then insert a rheostat upstream to reduce power and control temperature.

Someone else suggested a DC supply, but it wasn't clear whether this was a recommendation or a requirement.

I'd be grateful for any help I can get.

You said that the adapter was 40 va. Did you mean va (voltage amps)? Voltage Amps is another term for watts. Watts is voltage times amperage. At 24 volas 40 va is only 1.66 amps.

Yes, the cheap adapter I tried was 40VA (volt amps), and very insufficient for the heater.
It burned out (or blew its fuse) within a couple of minutes.

My problem is that I don't know how to keep the heater at a low enough temperature (about 105F) without lowering the amperage. And I'm not sure if I can use AC at all.

A restive circuit such as a heater shouldn't care whether the voltage is AC or DC. So a transformer from 120V to 24V and a rheostat should work.

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without lowering the amperage

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Lowering the amperage (rheostat) is how you control the temperature. In addition to the rheostat you should have some kind of maximum amperage limiting device as a safety device.

There is no DC transformer, you can't transform DC. All you need is a variable transformer, Variac is a trade name but not cheap. 120VAC to 24 VAC is a very common transformer as it is used in powering your home heaters, AC, thermostats, burglar alarms etc. All you need is one that can handle 10 Amps. BTW, that seems very high amperage. Google 24VAC, 40 VA transformer and there are loads or them online.