Do potatoes worsen Crohn's symptoms?

Potentially.  Here is the evidence I've found in support of eliminating potatoes to ameliorate Crohn's symptoms.  But what I've really learned, is that they should not be a part of any wellness diet, regardless of the specific underlying pathology.

Potatoes are polysaccharides.  Elaine Getschell, (Specific Carbohydrate Diet) among others, theorized that polysaccharides (starches) feed undesirable bacteria in the small bowel and large intestine causing IBD symptoms.  Since potatoes are predominantly starch (polysaccharide), digestion occurs in the large bowel.  Eliminating potatoes removes what is potentially a very significant source of polysaccharides, cutting off nourishment for harmful bacterias in a vulnerable gut.  See my post "What is the Specific Carbohydrate Diet?"

Potatoes are nightshades (solanaceae) containing  alkaloids, which are toxins, which may affect nerves, muscles, and joints in susceptible people.  Patients in the arthritis community commonly eliminate nightshades from their diets for symptom relief. See Dr. Childer's Arthritis Nightshades Research Foundation.  See my post entitled "Do foods from the nightshade family worsen the symptoms of CD?"

Potatoes contain glycoalkaloids (alpha-solanine and alpha-chaconine), natural defense mechanism, which makes the intestine even more permeable.  Peeling and high heat cooking does not eliminate glycoalkaloids.   To make matters worse, frying potatoes intensifies these molecules.  Diarrhea is among the effects of glycoalkaloid intake. Those with gut integrity issues are best to avoid worsening permeability.  See  my post entitled "Do glycoalkaloids worsen Crohn's symptoms?"  

Potatoes are a mainstay of the high carb SAD diet, which is associated with a plethora of negative health effects including inflammation, cancers, and metabolic syndrome. As a general rule, reducing or eliminating high carbohydrate foods is health-promoting.  

There is great interest in carbohydrate reduction in many wellness communities.  In the paleo community, see, for example, Mark's Daily Apple and Robb Wolf.  For an alternative wellness plans by medical doctors, see, for example, Dr. Mercola at Mercola.com and Drs. Eades and Eades at DrsEades.com.  For low glycemic foods see Atkins, South Beach Diet, and  Low Glycemic diets.

I cannot identify any downside to eliminating potatoes, other than dealing with the shock of others who find it astonishing that we are able to go on living without potatoes.  My kids don't miss them and neither do I.  We roast carrots and mash squash, and with all the other yummy food we're eating, we've never looked back.

Do nightshade-family foods worsen Crohn's symptoms?

Solanaceae at Wikipedia

"What are nightshades and in which foods are they found?" on the world's healthiest foods
"Potatoes, tomatoes, sweet and hot peppers, eggplant, tomatillos, tamarios, pepinos, pimentos, paprika, cayenne, and Tabasco sauce are classified as nightshade foods. A particular group of substances in these foods, called alkaloids, can impact nerve-muscle function and digestive function in animals and humans, and may also be able to compromise joint function. Because the amount of alkaloids is very low in nightshade foods when compared with other nightshade plants, health problems from nightshade foods may only occur in individuals who are especially sensitive to these alkaloid substances. Since cooking only lowers alkaloid content of nightshade foods by about 40-50%, highly sensitive individuals may want to avoid this category of food altogether, while non-sensitive individuals may be able to eat these foods, especially in cooked form, without problem.

...

Effect of steroid alkaloids on joint health ... Some researchers have speculated that nightshade alkaloids can contribute to excessive loss of calcium from bone and excessive depositing of calcium in soft tissue. For this reason, these researchers have recommended elimination of nightshade foods from the meal plans of all individuals with osteoarthritis, rheumatoid arthritis, or other joint problems like gout.

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Effect of nicotine alkaloid on health ... While we agree that the amount of nicotine in nightshade foods is very, very small, it still seems possible to us that some individuals might be particularly sensitive to the alkaloids found in nightshades, and that even very small amounts might compromise function in the bodies of these individuals."

Dr. Childer's Arthritis Nightshades Research Foundation

Nightshade Free