Solution:

The only triples of integers less than $10$ that multiply to $24$ are $$(1, 3, 8), (1, 4, 6), (2, 2, 6),$$$$\text{and } (2, 3, 4).$$

The triples with $3$ distinct numbers can be rearranged to form $6$ distinct $3$-digit positive integers. The other triple can be arranged to form $3$ distinct $3$-digit positive integers.

This leaves a total of $3 \cdot 6 + 3 = 21$ integers.

Thus, D is the correct answer.