Relation between torque and speed in a dc motor(say dc shunt motor)
At constant supply voltage, if you increase the load, that that slows the motor down, mechanically. The reduction in speed reduces the back emf, which leaves a higher voltage left over to push more current through the windings. The higher current generates more torque, which allows the motor to come to equilibrium with a higher torque into the higher load.

If you increase the current flowing through the motor by increasing the supply voltage, which increases the voltage across the winding resistance, that will increase the torque and the motor will speed up. Its load will probably require higher torque to drive it at the higher speed, so the motor will come to a new equilibrium at the higher speed, with a higer current and torque.