If a baseboard heater is leaking you can often tell by looking for any signs of water damage, decrease in heating performance and even a drop in water pressure with your heating system.
Look around the area of your baseboard heater for any signs of water stains or dampness near your baseboard heaters or even the boiler.
If you have a hot water baseboard heater system, check the water pressure gauge and if you notice a consistent drop in water pressure then it suggests you have a leak that needs to be fixed.
A baseboard heater that smells like burning plastic could be dust and debris accumulating on the heating element or something plastic may have gotten lodged in the baseboard heater.
Other reasons for a baseboard heater smelling like burnt plastic are faulty wiring, loose wiring connections overheating components and if it's a new baseboard heater the burning plastic smell could be the protective coating burning off.
If your baseboard heater is clean and free of any dust on the heating element it could be a loose or bad wiring if it's an electric baseboard heater or a bad connection or a wirenut is loose on the electric heater.
If the burning plastic smell goes away pretty quickly then the heater is likely safe but if the burning plastic smell persists with your baseboard heater you should shut it off and have it checked out to be safe.
It is normal for an electric heater and other heaters to smell burnt when you first turn it on after it has not been used for awhile due to dust.
When a heater sits unused for awhile it can accumulate dust and when you turn the heater on the heater can emit a burning smell due to the dust being burnt off.
This is normal and should go away within 10 minutes to 30 minutes or so but if not then it could be a sign that the heater is bad and needs repaired or replaced.
Even a brand new heater including electric heaters may smell burnt for a bit when first turned on due to protective oils and paint during the manufacturing process.
But the heater should not keep smelling burnt for too long or it's a sign that there's a problem which could result in a fire.
The burnt smell of the heater including electric heaters when just due to dust or the newness should fade away after a couple of minutes to a few hours and if not it could be a more serious issue.
A burning electrical smell is a more serious issue and can indicate overheating, damaged wiring or other electrical problems.
If you continue smelling burnt electrical smells it the heater should be turned off and unplugged or power cut off to it.
My electric baseboard heaters will emit a dust burning smell at the start of the season when I turn them on for heating as there can be dust that I cannot fully get out but it eventually burns off.