Many hotels, especially large corporate chain hotels, have an "express checkout" service where they'll put a copy of your bill under your door and invite you to simply leave without checking out if all is ok, with the balance of the charges billed to your credit card on file. Sometimes you can officially "check out" from your telephone or in-room TV. See In US hotels, what happens if you skip checkout and just leave the hotel?.
They certainly aren't going to think you've stayed for five years if you haven't checked out. At the end of your stay, they'll come to clean your room, notice you're gone, and take care of it for you. But some hotels, especially smaller hotels, do want you to check out.
So if you're unsure about the hotel's policy, want a complete copy of your bill, or just want the peace of mind that your business with the hotel is finished, there's absolutely no harm in stopping at the front desk to check out.