The dry season, roughly December to March, is the easiest time for Tiwai: roads are more reliable, forest trails are less muddy, and river travel is simpler. Wildlife is present year-round, but sightings depend more on patience and a good guide than on the calendar.
The rainy season brings lush forest and fewer visitors, but it also brings difficult roads, leeches, and the real possibility of delays. If you go then, build slack into the trip.