Can tree roots be killed after the tree is cut down?
Yes, but prepare for a war. Some trees fight harder to live than others. We cut down our windbreakers because their roots spread out 3 acres and were forcing the sprinker system to raise out of the ground - so our underground sprinkers each looked like they were on a little hill. After they were cut down, they fought like a war to live. You would see little trees springing up for 2 acres where the roots were and just had to keep mowing them down. Eventually the roots died, but it was war in the meantime. Be vigilant in getting the little ones that come up, if even on of them lives, the roots live.
Many roots you don't need to worry about. If the stump doesn't send up sprouts, the roots will die on their own as there will be no leaves to feed them. This is true of most conifers and some broad-leaved trees. Those that do sprout vary in how strongly they will sprout and thus how hard it might be to finally get rid of the tree. Most tree stumps (and thus eventually the roots) can be killed simply by cutting out the light to them for one to two years, usually by tying some thick black plastic (such as a thick plastic bag) over them and flaring it out a few inches. If the sprouts poke through the plastic anyway, just use a thicker layer. This works for trees that send up sprouts only from the trunk. Some tree species also send up sprouts also from the roots, such as Plum trees, or the tree mentioned by the other answerer. These are harder to fight. Even having the stump ground out won't get rid of them. The easiest way then, is to paint undiluted roundup on the stump. (Read the label. Roundup is not effective when used in cold seasons) If you do this, don't let the roundup run into the soil, as you might well kill some plant you didn't want to kill that happened to have roots in that area. Some sprouts might well come up anyway from roots and when they do just treat them as you did the tree stump: cut them down and paint roundup on them.
If you don't want to use chemicals, you are stuck with religiously pulling the new shoots as they appear. Another way, however, is possible, which is to create an impenetrable barrier to the sprouts. A great way to do this is to spread out many overlapping layers of newspaper (at least 6 or more) on the bare ground and then cover this with mulch. Eventually the newpaper breaks down and becomes part of the soil, but by then the stump will mostly likely have given up. Also, as the previous answerer alluded, if they are in a lawn, you can just keep mowing them until they give up. The trick is to keep them from forming leaves so they can't feed the roots.