You might be able to ID exactly which Dieffenbachia it is, if I'm guessing right about what Dumb cane is.
Dumbcane (Dieffenbachia 'Tropic Marianne')
I think it's hard to know how harmful the thing is, unless someone recognizes exactly what it is. The consensus seems to be "maybe a fungus, mold or mushroom". I think that many would not bother most plants, but is soil that encourages mushrooms really the best kind of soil for a potted plant?
If it is any kind of mold, fungus or mushroom, it can't hurt to remove as much of it as you can without hurting many roots. In my yard, burying wood chips under the soil caused a lot of fungus. But the Pacific Northwest and NJ are two very different climates, so you might have some different cause. What do you fertilize with? If you do a lot of spot composting, and that mushroom (or whatever) is growing right out of some buried compost, maybe you should compost scraps in a separate pile before burying them near the plants.
I hope you write back to let us know what it is, or with more details.
My guess is that, if it is any kind of mold, fungus or mushroom, it showed up because it had lots of organic matter to eat and favorable conditions. If you starve it, and maybe water it less, it will get hungry and at least the above-ground part will disappear on its own.
Can the Dumb Cane survive a frost? If not, I guess it's an indoor plant.
One possible alternative to dunking the whole rootball would be to eplace as much soil as convenient, like re-potting.
Another, easier alternative is to water with rather dilute Hydrogen Peroxide for a while. Dilute enough, it won't hurt the roots. You can also spray leaves with it, if they seem to have a fungus infection
Start with "drugstore peroxide" which is 3%.
Then dilute it either 1/32 or 1/16 to get 0.1% or 0.2%.
Many people use it stronger, so these are very safe concentrations.
For 0.1% H2O2:
1 ounce of peroxide per Quart of water
1/2 cup of peroxide per Gallon
For 0.2% H2O2:
2 ounces of peroxide per Quart of water
1 cup of peroxide per Gallon