How long should I run my dog in hot weather to avoid heat stroke?

You are a great pet owner for being concerned about this and planning ahead! In the oppressive heat, heat stroke can happen very quickly. Some dogs can learn to adapt to the heat and are better able to handle it (think dogs that work as partners for soldiers in Iraq or Search and Rescue dogs).

Signs of heatstroke include:

Increased heart rate
Excessive panting
Increased salivation
Bright red tongue
Red or pale gums
Thick, sticky saliva
Depression
Weakness
Dizziness
Vomiting (sometimes with blood)
Diarrhea
As heatstroke progresses, it can cause seizures, coma, cardiac arrest, and death.

Exercising early in the morning during the cooler hours, providing lots of fresh cool water and finding places to exercise in the shade (like a nature trail hike) can help keep your dog cool. Several companies make Cool Coats that reflect heat and funnel it away from your dog.
Also, find other ways for your dog to get exercise or even exercise their mind. You would be amazed at how tiring this is for dogs- take a Nosework class and your dog can learn to find different scents and you can practice at home in the house. Pick up a few puzzle toys and let your dog learn to find food and treats in the puzzles.
Also, you could see if your dog might enjoy a swim-- even a little splash pool at home can be fun for the dog or some hose play- Don't overdo a dog's can get very sick if they drink too much water when playing.