Hi Marlene: If you're talking about hard-shelled nuts, like walnuts, then, no. You don't eat the shell, and what is inside the nut grows in a sealed, sterile environment. On the other hand, nuts where you put the whole nut in your mouth, like sunflower seeds, to crack them open - those I would wash (if they're not organically grown).
It is something to think about. Whatever is on the outside of nuts and other foods that you just eat the inside of could get on your hands and be transferred to the contents that you then touch and put in your mouth. There are fruit and vegetable washes that are able to remove pesticide residues, produce wax, and dirt. Just spray it on and rinse it off. Just a little precaution :-)