Another example: At our spa, we use a special herbal wrap to help you work up a sweat to detox.

Another example: Apply the shampoo to wet hair, and then work up a good lather.

"To work up" is a phrasal verb which is defined as "to develop", irrespective of the means. It does not have to be "doing work". You could work up a sweat by sitting in a sauna.

In this case you are doing work (lifting weights, or building a house) and it is causing you to sweat. Here are a couple examples: "I am working up a sweat lifting all these heavy rocks" or "I like going to the gym (or health club, or fitness club) and working up a sweat." In most cases when we use 'working up a sweat' the focus is on the activity we are doing, not the actual sweating. We aren't bragging about how active our sweat glands are, we are pointing out that we are doing something difficult (hard work, or exercise).