No it does not. You wet your shirt with unheated tap water, wring it out, and wear it. Repeat as necessary every 15 minutes to 1 hour in extreme heat, no matter how humid it is.

In high humidity by wearing a shirt doused in tap water you cool down by heating up your t-shirt. This method does not work if water comes out of your tap already hot. This only happens rarely. In dry environments evaporative cooling helps but it is not necessary to be effective.

I recommend using shirts made of elastic breathable material such as under armor shirts for comfort. Cotton tends to stick to your skin extra and feel really nasty.