What should I do if my MacBook Pro smells like burning?

I would go a bit further. If your computer smells like it's burning, turn it off, unplug it and contact Apple to arrange to have them look at it:

Genius Bar Reservation and Apple Support Options - Apple

Don't use it again until they do.

Without a qualified technician there’s nothing you can do yourself. If the smell is like something is burning then stop using it immediately until you can take it to an Apple Store or authorized Apple service provider for examination and diagnosis.

If it just smells hot then, as AlWeir points out, your fan intakes might be clogged. Buy a can of compressed air for cleaning computers (available in any electronics store) and try blowing it through the air vents. You could also try using a standard vacuum cleaner and nozzle to do the same.

In either case you must do something. Your MacBook shut down because it overheated. If allowed to continue to overheat your MacBook will become damaged beyond repair.

I should add that I had that smell once and it coincided with the base of my Mac expanding and on a flat surface it would minutely teeter-tooter. The cause was from expanded batteries. Thankfully I had a backup of the computer in a SSD and was able to use on a purchase of a M1.

AlWeir wrote:

I should add that I had that smell once and it coincided with the base of my Mac expanding and on a flat surface it would minutely teeter-tooter. The cause was from expanded batteries. Thankfully I had a backup of the computer in a SSD and was able to use on a purchase of a M1.

Exactly the sort of situation I was envisioning.

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It might be your fans are clogged.