In June, some rain can be expected in Bali and Lombok and much of Indonesia once every 4 or 5 days. June is a great time to travel to Indonesia.
Sumbawa, Flores and Komodo Island have climates similar to Bali.
The islands of Nusa Tenggara should see long clear days of sunshine and little rainfall.
By June, the dry season in Java, Sumatra and Sulawesi is well established. The occasional afternoon shower may be expected but generally expect long clear days of sunshine.