Why do some fruits and nuts make my tongue itch? When I was young,
walnuts would make my tongue "itch" for a lack of a better
description.

Over the years the list has grown (I'm 40 years old now) to include
walnuts, pecans, almonds, cherries, apples (any/all), plums, pears,
peaches, eggplant, and others.

I have notice that if the fruits are cooked or processed (apple pie,
baked apple, apple juice, cider, etc.) any side effects vanish.
However, peeling free fruit, buying oraganic, etc. seems to have no
effect.

Any idea what's causing this, and other than cooking my fruit, any way to avoid it?

It sounds like you are suffering from oral allergy syndrome.  Check
out the following links:

http://www.inspection.gc.ca/english/corpaffr/foodfacts/orale.shtml
http://allergies.about.com/cs/oas/a/aa052499.htm
http://www.calgaryallergy.ca/Articles/Oral_Food_Allergy.htm

JR