Does the required torque to spin a propeller increase with RPM?
Of course. More and bigger blades is more drag. Drag is also proportional to the velocity squared. Even maintaining the RPM, a bigger propeller (diameter) will have the additional diameter flying faster.

To maintain a higher RPM (or to use a bigger prop) the force to cancel the increased drag is higher, and thus the torque (force times distance) that drives the propeller.

Place the propeller in vacuum, and the engine will have no trouble with the propeller apart from any loads on the shaft (but there won't be any thrust).