Burnt plastic smell is harmful especially if you breathe in burnt plastic smells too much.
Burnt plastic creates and releases toxic fumes into the air which can also be carcinogenic.
The specific bad chemicals that the toxic fumes release when plastic burns are often dioxins, phthalates and other various volatile organic compounds.
The fumes and smell from burning plastic can irritate your nose, eyes and throat and in more severe cases the burnt plastic smell and fumes can result in more serious health issues such as neurological damage and respiratory problems including lung cancer, COPD etc.
Breathing in a small amount of burnt plastic smell and fumes is unlikely to cause any serious issues but over time it could result in health issues especially if you're constantly breathing in burnt plastic fumes.
If you're in a building or home that has a burnt plastic smell you should open any doors and windows to allow the place to vent and help bring in fresher air.
Also go outside and get some fresh air to help your lungs recover.
If you know where the burnt plastic smell is coming from such as an appliance or other electronics etc turn whatever it is off or if it's electrical connection turn the breaker off.
Don't make a habit of breathing in burnt plastic smells but you're likely gonna be okay if it just happens once or twice or occasionally.