Mom's face was red like a sunburn and very warm. Does anyone know why this happens?

Review the side effects of the meds she's on. Review supplements (Niacin, which is B5, as mentioned below can cause flushing). Other thoughts: thyroid? endocrine issue? It an be a sign of a serious underlying condition so do contact her doctor.

Hot flashes?

For example, taking Niacin can make a person flush red in the face.

Just waking up from a nap where one was too warm.

After showering, one can flush red.

Do you know if she's ever had rosacea? I have that, and it causes my face to flush/break out periodically. It does cause marked redness. I take antibiotics and use topical medication for it. It can be triggered by different things, such as heat, cold, spicy foods, hormones, etc. I would keep track of when it happens, how often, etc, and see if you notice any patterns (what she ate, climate temperature, etc), and let her doctor know.

Good idea to get her vitals when this happens,   My first question before finishing your post was whether or not there were manifestations in vital signs.  Good thinking!

Is this the first time it's happened?  If not, is at certain times, following activity, or something else?  I'm thinking that perhaps something is preceding this but you'll have to try to isolate her activities to determine that.  

I also wondered about food allergies.

Another issue is what meds she's taking, any new meds, and/or whether there might be interactions if a new med was introduced to her regimen.

Red wine is notorious for this flush reaction. It can be any allergic reaction, even a reaction to going out in the cold. If it is full body it bears instant action as can be a full body reaction to something if it doesn't fade quickly. My Mom reacted to coumadin, a rare and life threatening whole body "burn" that was dangerous.

Food allergy? If this happens again, ask Mom what she had to eat recently. There might be a pattern.

The only time I have seen this was a fellow who overdosed on vitamin E. Not dangerous, just turned bright red...like a sunburn...and the skin was hot.

doctors said that the fairer your skin, the less vitamin E it takes to get this effect.

Musame, medical emergencies do not always involve pain. My husband had serious heart issues and before that he had a stroke and was not in pain with either incident. You need to report this to her doctor as soon as possible. They can advise you how to proceed.

There are many and varied reasons Musame.
I am going to let others answer, but I wanted to acknowledge your question one hour ago. There will be answers to come, even though it is getting late on the West Coast of the USA.

My father had that happen, it was called a vasomotor instability, and was quite normal for him his entire life.