Probably Not. Skin to skin contact usually doesn't cause a rash, so it is doubtful that picking up your child would cause a rash. Babies occasionally can have fungal infections under their armpits, given the moist environment. If your baby has a rash under their armpits and it is not improving, you should consult your physician.
It's possible. If the rash occurs in the same locations each day after you pick him up it is likely that his skin is sensitive and your handling may be the cause of the rash. Check the ingredients of any hand lotions you may be using as a moisturizer. Some infants have very sensitive skin and react to fragrances and/or other ingredients. If the rash worsens, spreads or has blisters it may be an infection.