When is the best time of year for riding on Naxos?

On the road to Potamia, December 17, 2015, about 14°C according to my GPS.
Short answer: Avoid coming to Naxos in July and August if you have a choice. Riding is best in spring and autumn.
July/August is the tourist main season, everybody is here. Which means a lot of cars, buses, scooters, 4-wheel-bikes on the road. Which also means accommodation is going to be more expensive. And it's hot. You can ride in the heat, but it's less pleasant and you better take it easy, take precautions, and do a bit of acclimatization. Why bother? Come in spring or autumn. Even our "winter" is pretty much rideable.
But let's look at it in detail!