A fireman is sliding down a fire pole. as he speeds up, he tightens his grip on the pole, thus increasing the vertical frictional force that the pole exerts on the fireman. when the force on his hands equals his weight, what happens to the fireman?

According to Newton's first law, when the net force on the body is zero, the result will be that the body will have constant velocity. So when the force of friction on fireman's hands and the weight are equal, the fireman will slide down the pole with a constant velocity.