For example, would a box of baking soda in a fridge lose weight as it absorbs odor? How long would it take to drop 100 g?
Let us assume that baking soda works by not simply absorbing, but reacting with acidic/odiferous substances via an acid-base reaction. So, for example if you had rancid butter in the fridge and it gave off the nasty-smelling butyric acid, that acid would react with the baking soda to give sodium butyrate plus carbonic acid (which breaks down to carbon dioxide and water). Let us assume that the carbon dioxide and water would evaporate. That would be a net gain of weight since the butyric acid molecule has a bigger mass that carbon dioxide and water.
Most odiferous substances will likely have a higher molecular weight than carbon dioxide plus water, so I believe over time the baking soda will increase in weight. This will also be true for those cases where there is no reaction - but simply absorption.