Yes, you can stay over and it is a great idea as the island is lovely in the evenings once all the day-trippers have left. If you'd like to stay over then your best option is to stay at a guest house, there are no hotels on Burano itself. The best guest house option is Casa Burano. They offer rooms in beautifully restored fisherman's cottages and their cottages are to be found at various locations across Burano. All offer a high standard of accommodation, free WiFi, breakfast, 24 hour coffee, bathrobes, slippers and air conditioning.