Can a whitening toothpaste with baking soda irritate ones gums?

Yes. Any toothpaste can irritate the gums baking soda in te mouth has an increase abrasive products.
Yes. Usually it isn't a problem, but if you are experiencing irritation your gums can be irritated by: the abrasive action of the baking soda, the peroxide that is used to whiten the teeth, the sodium laurel sulfate (detergent), and sometimes by the flavoring in the toothpaste.
Sometimes. Could be the whitening agent and/or the baking soda. Also, make sure to be using a soft toothbrush, and only brush with it twice daily. The active ingredient is hydrogen peroxide. Safe and effective, 1 part 3% otc hydrogen peroxide to 2 parts water after brushing twice daily for great effects 4-6 weeks. Switch to sensodyne toothpaste for a while before whitening again.