Sweat dries faster than water most likely because it has less water per drop, the rest is dissolved salts,,, which do not evaporate (they just dry out).
There's actually very little dissolved salt in sweat - it's 99percent+ water - so I doubt it makes much difference. In fact, I'd be very surprised if sweat evaporated at a significantly different rate than pure water.
I think the difference Andy is observing is probably because the sweaty shirts aren't actually as fully soaked with water as shirts that have been washed in a bucket of water. In other words, the sweaty shirts really are drying faster, but that's because there's less water in the fabric to evaporate.
I have been considering that even though these shirts appears sweat soak, they really are not a wet a the washed shirts, maybe I am just fooled. I am taking a small pinch of sea salt they sell in the open air markets here, because I am getting worried about the amount of salt I am losing due to sweat.