WHY? If they are wearing a thick coat, the padding in the coat will not allow you to snug up the harness correctly and the harness will actually be loose on the child. The padding in the coat will compress during a crash until it gets to the body which doesn't allow the child to ride the crash down appropriately.
Important to note: Car seat manufacturers cannot determine if it is safe to use a thick winter coat because they do not crash test with coats on the test dummies. They do test for misuse and they can determine if a loose harness causes injury.
DO: Follow the car seat manual for proper harness fit, installation and use. The manuals are written based on crash test data.
DO: Use an appropriate car seat no matter what the weather conditions are outside.
DO: If there are no safety issues in doing so, heat up the car ahead of time.
DO: Put a thin fleece jacket on your baby, toddler or older child.
DO: Put a toboggan hat and gloves on your child. This definitely helps retain some internal warmth.
DO: For a newborn baby, many infant carrier car seats come with a "warming boot" to put over the baby. If not, one may be purchased from the manufacturer of the car seat or they may have approved the use of one manufactured by another company.
DO: Throw a blanket around your child when carrying them out to the car to keep them warm. (If you are super nice, you might have even warmed it up in the dryer first.)
DO: Place them in the harness and snug it up. Then you can place the blanket over them so it doesn't interfere with the harness and tuck it in around the child's sides. To avoid the possibility of suffocation do not put the blanket near their head.