Is it really necessary to trim the small branches off of the small limbs before wood chipping them?

Yes, it is true, they spring out as they go down the chute into the chipper and clog the chute. You then have to shut the chipper down and remove the clog. If you don't shut the chipper down and the clog comes free while you have your hand in the chute, you could be sucked into the chute and loose an arm.