http://articles.latimes.com/2008/jan/28/health/he-skeptic28 :

The bottom line: Sweat does contain trace amounts of toxins, says Dr. Dee Anna Glaser, a professor of dermatology at St. Louis University and founding member of the International Hyperhidrosis Society, a medical group dedicated to the study and treatment of heavy sweating.

But, Glaser, adds, in the big picture, sweat has only one function: Cooling you down when you overheat. "Sweating for the sake of sweating has no benefits," she says. "Sweating heavily is not going to release a lot of toxins."

In fact, Glaser says, heavy sweating can impair your body's natural detoxification system. As she explains, the liver and kidneys -- not the sweat glands -- are the organs we count on to filter toxins from our blood. If you don't drink enough water to compensate for a good sweat, dehydration could stress the kidneys and keep them from doing their job. "If you're not careful, heavy sweating can be a bad thing," she says.

Sweating definitely won't help clear the body of mercury or other metals, says Donald Smith, a professor of environmental toxicology at UC Santa Cruz, who studies treatments for metal poisoning. Almost all toxic metals in the body are excreted through urine or feces, he says. And less than 1% are lost through sweat. In other words, you'll do far more detoxifying in the bathroom than you ever could in a sauna.

http://www.walkaboutmag.com/17gates.html :

I’m always stunned when people attempt a variety of strategies to “clean their body of toxins.” Most of these practices sound logical, but body metabolism, which is sophisticated and complicated, is not a logical process. There are no toxins in sweat. Toxins are removed by a healthy kidney and liver working full time to counter all we do to abuse our bodies with poor exercise, eating, and lifestyle habits. The body isn’t detoxified when it sweats. It’s overheated.

http://getbetterhealth.com/toxins-can-you-sweat-them-out/2010.09.22 :

You might not want to believe me, but it’s fact: You cannot sweat out toxins. Sweat is composed of 99 percent water and a tiny percent of salt, urea, proteins and carbohydrates. Salt, proteins and carbohydrates are natural. Urea is a by-product of protein metabolism and is non-toxic. It’s regulated to keep your blood at a healthy pH. Most excess urea is eliminated in urine (hence the name) and a small amount is in sweat.

Toxins like mercury, chemicals, alcohol, drugs, and spicy BBQ sauce are eliminated by your liver and intestines. Sweat glands, all 2.6 million of them, regulate your temperature — they’re not designed to expel toxins.

The primary ingredient in sweat is pure water. The water evaporates from your skin, cooling you. Excess sweating doesn’t eliminate excess salt or help hangovers. By forcing your body to copiously perspire, you’re only forcing your kidneys to save water (and ironically actual toxins) elsewhere. The water that ends up in the towel on the floor is the precious water you needed to stay hydrated, not a puddle of poison.

In some ways sweat is the opposite of toxic, it’s a vital fluid. When you are working out hard, replace it. I recommend water, not Captain Morgan rum.

Exercise makes a headache in progress worse, and you can see I don't have that many windows of headache-free-ness. That said, if some modest amount of exercise would really help, it might be a net positive trade to make. That said, [citation needed].