What causes hair extensions to sometimes have an unpleasant odor?

Other reasons why hair extensions might smell like mildew include:

Yeast

Malassezia yeast eats sebum, which is found in dirt, sweat, styling products, and a film that protects the yeast from being washed away.

Bacteria or fungal overgrowth

The scalp has a natural microbiome of bacteria and fungi, but an overgrowth can cause an unpleasant odor.

Product buildup

Using too much hair product or not rinsing it out thoroughly can cause buildup on the scalp.

Diet

High sugar and high-fat foods can alter your sweat, which combines with bacteria on your skin to create a foul odor.

Air pollution

Smelly particles from air pollution can cling to your hair and scalp.