In some healthy people, furry tongue appears to be a normal condition of unknown cause. Because it can be something of a hygiene problem, I recommend regular tongue brushing and scraping (with a downward-bent, commercially available scraper). Scrape as far back as possible. You'll gag at first, but this reflex is quickly overcome. Gargling with bicarbonate of soda -- as already suggested -- helps hygiene. Or use dilute (1%) hydrogen peroxide (but peroxide may aggravate the condition in some people). None of this cures the problem, but it does control the condition with little effort.
FURRY TONGUE? USE BIO CARBONATE OF SODA AVAILABLE IN BOOTS. HUGE BOX FOR ABOUT £2.00. BRUSH EVERY MORNING AND NIGHT IF YOU WISH WITH THE BIOCARB ON THE TOOTHBRUSH. START RIGHT AT THE BACK OF YOUR TONGUE YOU MAY WANT TO MAKE SICK SOUNDS BUT YOU WILL GET USED TO IT. I LIVE WITH A NICE PINK TONGUE NOW. YOU CAN ALSO USE BIOCARB DILUTED IN WARM WATER IF YOU HAVE HEARTBURN AFTER EATING OR DRINKING TOO MUCH!