No, he won't bite. He just finds running much more fun than walking.
I was just starting up the bridleway to the Outwoods when I passed a lady with a dog, and the dog started running around my feet, as so many dogs do, not quite tripping me up. The lady called the dog back, which is usually the end of the encounter. Indeed, I thought it was the end of the encounter this time, but a little further up the path I became aware of panting behind me. It stayed just behind me, not threatening to trip me up. This continued for maybe a quarter of a mile, until I stopped, turned round and shouted "Back", pointing back down the path. I had to do this twice before the dog got the message.
The only similar incident I have had was when I was living in Bracknell, and running in Crowthorne Woods. A Great Dane detached from its owners and came bounding along beside me. After maybe 200 metres I altered course to make a loop back to where the owners were. They apologised and told me that he was only a puppy.
The dogs in both incidents were lovely animals. So the question is: if a dog follows me all the way home, is it mine to keep?