Ah yes, the rubber "snakes" which can get picked up. I have experience with these autocrossing on grippier compound tires. For best grip, yes, you'll want to remove them. It's easiest to do it in between runs (when they are still warm), as in that case you can use just some rough gloves to do it. If you let them cool down and "harden", you may need some scraping tools.
When changing tires back to street tires after an event, I routinely scrape and wipe all all the debris on the outside of the tires (rubber snakes and embedded gravel) on racing tires before wrapping them up and putting them away into tire bags.
Usually the snakes get picked up driving the car back to the grid/paddock after a run. If you don't care about optimizing grip (e.g., for intense competition), sure, you can just drive with them. They could prolong treadwear!
Removing gravel from racing tires is also a lot easier on a warmed up tire than after they've cooled!
By the way, I would not recommend using a heat gun, if you want to maximize the life of the compound. Heating it up more often than necessary can make it get harder quicker. Wrapping them into tire bags and storing them in a cool environment (definitely out of the sun) will help them last as long as possible.