For some reason people and animals can't resist poking at exposed speaker cones. The usual victim is the dust cap. I'll go over the procedure for replacing dust caps some other time.
Its not that easy to rip through a paper speaker cone but it happens. One way to do it is when you use an electric screw gun or drill to mount the speaker and it slips. Oops, a hole in the cone. Another way is when a heavy sharp edged object accidently falls on the cone. Ok, so you have a rip, tear or hole and you want to fix it. As long as the voice coil is ok and doesn't rub in the gap, there's no reason to recone the speaker and a repair is pretty easy.
What I do is get some tissue paper and rubber cement. Around Halloween time its easier to find black tissue paper, which blends in better than white on the typical speaker cone. The very common rubber cement that you find everywhere they sell school supplies, is perfect for the job and it comes with a brush, too, which is perfect for applying the cement.
First, remove the speaker from the enclosure so you can get at the front and the back of the cone. With one hand in the back and the other on the front use your fingers to press the paper back to its original position. Try to mess the fibers at the edges together again. The cones are usually made from wet pulp on a screen and then pressed for stiffness, so you're just trying to press them back where they were. Then I go into the back of the speaker to apply a patch to reinforce the damaged area. A 1 inch wide strip of tissue paper extending about a half inch past the ends of the rip will be plenty enough and you could go down to a half inch or so. If there's a big hole with missing pieces you could patch it with a similar type of paper as the cone, but it will never look good, and the sound of the higher frequencies will be changed. Heavy patches on a subwoofer speaker won't be heard but keep in mind that any time you add weight to a cone you change the resonance a little bit, so use minimal material and adhesive. As for the repair: Apply a thin layer of cement to the back of the cone over the damage and extending to where the tissue paper will cover. Note: its sometimes difficult to access the damaged area if its right behind the basket structure but the brush will usually reach. Don't worry too much about making a bit of a mess because its on the back. Wipe extra cement off or brush it in evenly. Speed is important because the cement will start to dry. Position the tissue paper patch over the rip and gently press it onto the cement from both sides of the cone as before. Some cement may ooze out the front and that's good, just wipe it into the tear quickly and move to the back. Now brush cement onto the patch from the center to the edges smoothing out any wrinkles. If its hard to access get it from another angle or put the patch on the front side if you don't mind the look. In some cases where the speaker is very difficult to remove and there's a grill cover to hide the speaker I will do the repair from the front only, but that is not the best way, mainly because you can't press the fibers together from the front only. So now you have a patch. After an hour or so you can check the repair and if you think it could use a little more strength then apply another layer of rubber cement. The tissue paper weighs almost nothing and the rubber cement is not going to raise the total mass of the cone by a significant amount unless the speaker is small and you use more cement than is required.
By the way, if you're in a hurry and you don't have time to go get tissue paper and rubber cement you can use any paper with fibers like a paper towel or even toilet paper or a used dryer sheet, which is very strong but I'm not sure if the chemicals left behind on the dryer sheets will eventually react with the adhesive. So far I haven't seen it fail. If you don't have rubber cement you can use common white glue because it does stay flexible enough if you apply it in thin layers. Use a little paint brush. Any all purpose style glue except cyanoacrylate (super glue) will probably work in a pinch.
So basicly what you do to fix a ripped speaker cone is glue a paper patch over it.