What is the relationship between field strength and efficiency in a DC motor?
To a first order approximation, none. 

Modifying field strength is used as a speed control on some very large DC motors, and they don't dissipate huge powers and catch fire at either end of the useful range.

What it does do is modify the generator EMF at a given speed. Reduce the field and the generator EMF reduces. In a motor, that's called back EMF, so the supply voltage isn't cancelled by back EMF, current increases, torque increases, motor speeds up, back EMF increases, current decreases to the correct level to maintain the new speed.

To a second order approximation, there will be effects on efficiency.

Too much field will saturate the magnetic circuit. I'm not sure how that affects efficiency in a PM motor, but it certainly wastes power driving the field winding in a wound field motor.

Too little field attempts to run the motor very fast, but with little torque. Unless the motor is correctly (very lightly) loaded, it will tend towards stall, which definitely reduces efficiency. And ultimately, bearing/brush/windage friction will consume all the mechanical power leaving none at the shaft.

So, loosely, high field is generally more efficient.

But if the motor is correctly rated and not overloaded, there should be a reasonably wide range of field strengths at close to peak efficiency.

(If you have a fixed field, you can achieve a similar variation in results by winding the motor with more or fewer turns : more turns being roughly equivalent to higher field. If you don't feel like rewinding motors to experiment, radio control model car suppliers sell the same motor wound with different turns for performance tuning)