They can start growing again in the compost. That in itself isn't a big deal, since you an always pull up the plants, shred them, and recompost them.
The more serious issue is that you may be infecting the compost with assorted fungus and virus diseases, Potato blight is the most obvious risk but potatoes can also carry viruses that affect related species, e.g. tomatoes. Viruses will never be killed by composting - if you want to recycle the organic waste, burn it and use the ash. If you get potato blight fungus in your soil, it will take years to eradicate it.
Note, the diseases are carried in the leaves and stems, not just in the edible bits of the plant! Composting kitchen waste containing cooked potatoes is unlikely to be a problem.