First, you should have some field recordings to use as a reference, at least. Depending on the verisimilitude you are looking for, you might need to really hone in on the exact type of forest that is going to appear on screen. For example, a tropical rain forest is very different from a northern coniferous forest.
If you take the wildlife out of the sounds of a forest, then you're basically left with trees and undergrowth. Undergrowth may not make a lot of noise if there's nothing to step on it or walk through it. Trees, on the other hand, only need a little bit of wind or water to make noticable noise.
I've been told it rains every day in the rain forest, and maybe multiple times a day in places. So you may have water still dripping down through the canopy for a while after the rains, even if it's not raining on screen. Also, any amount of wind will stir thousands to millions of leaves, and even if only one leaf makes a very small sound, all those leaves together can make quite a racket.
A water drop falling on a leaf is a pretty flat sound. Like a small splat. Almost a pulse wave with an almost instant decay would work. Just like a very short pop with the low end filtered out. Again, actual recordings as reference would be good here.
Wind in the leaves definitely has a white noise component, but it's got a lot of character. You might have to layer up something more rattle-like with filtered noise. Again, not a lot of low end so high pass filtering will help.
Not really part of the soundscape but part of the overall sound design: there can be significant reflections in the form of echoes and reverb from the trunks of trees. This will vary a lot by the type of forest, with some sounding more dead and others sounding more live.
If you haven't already had a chance to research exactly what sounds you're going for (perhaps look for documentaries filmed in the same environment you want to re-create), I think that will be very helpful in terms of synthesizing a convincing sound. At the same time, be careful of being too critical. You just need enough realism for the viewer to fill in the last 10% or so. The sound will be background and the viewers will be expecting to hear forest sounds since they will be seeing forest on the screen, so their brains will help fill in the cracks if you can get them 90% of the way there.