It’s actually best to keep your balance transfer card after paying off your balance, even if you decide not to use it. Closing an account shortens the length of your credit history, which can negatively impact your credit score. It will also affect your credit utilization ratio, since the amount of available credit you have access to decreases. And if your new card has perks, like extended warranty coverage, or offers rewards, you can still enjoy these benefits even after you pay off your balance.
However, if your balance transfer card has an annual fee you can’t offset every year or you are worried about overspending with multiple credit cards, then closing the account might be the better option.