Choice $D$ is correct. The cost of the rental fee depends on the number of hours the surfboard is rented. Multiplying $t$ hours by $10$ dollars per hour yields a rental fee of $10t$ dollars. The total cost of the rental consists of the rental fee plus the $25$ dollar service fee, which yields a total cost of $25+10t$ dollars. Since the person intends to spend a maximum of $75$ dollars to rent the surfboard, the total cost must be at most $75$ dollars. Therefore, the inequality $25 + 10t \le 75$ represents this situation.

Choice $A$ is incorrect. This represents a situation where the rental fee, not the total cost, is at most $75$ dollars.

Choice $B$ is incorrect and may result from conceptual or calculation errors.

Choice $C$ is incorrect and may result from conceptual or calculation errors.