When $N$ is divided by 10, 11, or 12, the remainder is 7. This means that $M=N-7$ is divisible by each of 10, 11, and 12. Since $M$ is divisible by each of 10, 11, and 12, then $M$ is divisible by the least common multiple of 10, 11, and 12. Since $10=2 \times 5, 12=2 \times 2 \times 3$, and 11 is prime, then the least common multiple of 10, 11, and 12 is $2 \times 2 \times 3 \times 5 \times 11=660$. Since $M$ is divisible by 660 and $N=M+7$ is a three-digit positive integer, then $M$ must equal 660. Therefore, $N=M+7=667$, and so the sum of the digits of $N$ is $6+6+7=19$.