Burning sensation on edge of MacBook Pro

Most likely there is an issue with your home's electrical wiring (or possibly with a power/surge strip). You should have an electrician check out the home's wiring. You can perform a basic check of the home wiring by using a Receptacle Tester such as this one which will detect some common types of wiring problems...if any are detected, then have a certified electrician fix it for you.

https://www.kleintools.com/catalog/electrical-testers/receptacle-tester

You can use a grounded Apple Power Extension Cable (aka AC Cord) which should prevent the electrical tingling. The default Duckhead adapter provided by Apple for the charger is not grounded. Unfortunately not all regions have a grounded three prong Power Extension Cable available (they may have the cable, but it may not have three prongs). Here is the US version:

Power Adapter Extension Cable - Apple

You can also take the laptop, charger and charging cable to Apple or an Apple Authorized Service Provider to have them examine the items to confirm they are Ok.

Then it won't make any difference if there is no grounding pin.

If you connect any external devices which are connected to power using their own dedicated power adapters, then one of them may be at fault.

You should have Apple or an AASP examine the laptop, charger & charging cable just to make sure the laptop is Ok. Perhaps there is another reason for the sensation you are feeling which is not an electrical issue, such as some type of allergy to the device itself....I did see one other user mention some type of allergic reaction.