What is the exact formula relating a DC motor's input voltage and torque produced?
$$T=K_t\cdot i$$

Torque is proportional to the current, the torque constant \$K_t [Nm/A]\$ specifies the relation .

There is no direct formula for the voltage relation.







simulate this circuit &ndash; Schematic created using CircuitLab

Voltage to current relation, with inductivity neglected 

$$V_a = R_a\cdot i_a+\Omega\cdot K_v $$

Voltage to current relation, with inductivity into account
$$V_a = R_a\cdot i_a + L_a\dfrac{di_a}{dt} + \Omega\cdot K_v $$

You have also a transfer function, if it can help : https://electronics.stackexchange.com/a/262069/82111