This is a common practice among more devout Muslims. I’ve tried it, and it actually does work, though if you chew too much you’ll wind up with a bunch of little wooden shards in your mouth.
If your mouth is tingling after you eat carrots, you might want to stop eating them and have yourself tested to see if you’re allergic to them. Carrot allergies are one of the more common food allergies, and tingling mouth/lips is a common reaction to eating them.