The chewing gum itself is not what kills a dog when ingested but instead it's the sweetener used in chewing gum that contains Xylitol.
The Xylitol is what is actually harmful to the dog so you should avoid letting your dog get into chewing gum and never give the dog any chewing gum because it can be potentially fatal.
If your dog does it chewing gum the dog needs to get medical attention to prevent low blood sugar, seizures and even liver damage in your dog.
If the dog swallowed some chewing gum take them to the vet just to be safe.