Will my camera be okay in the rainforest?

I have a Nikon D40X with a standard Nikon 18-55mm lens and two Sigma lenses (telephoto and a macro). I'm going to the Amazon rainforest in Ecuador for 3 weeks and we have been told "nothing will dry, bring dry bags, you'll be damp all the time", that sort of thing. Will the dampness get into my camera or lenses? Should I take precautions? I'll obviously have cases for them all but I'll probably have them out for long periods of time. I'll try to take at least two of my lenses.

You should probably leave the D40 at home - Buy a Coolpix S30 or AW100 and you'll be set.

Get yourself a zip lock bag and keep your cameras in it since it will be raining most of the time. And put the lens and camera into a dry box when you are back from the trip.

None of those are designed to be used in that kind of environment. Leave them home.
I'd look for a cheap point & shoot with a zoom lens.

I would take a simple waterproof camera or even 2, something branded as 10 metre water resistant. Leave interchangeable lens gear at home unless you a professional and can write the gear off as a business expense. If you get say 12 megapixel you can do digital blow-ups later. Think about how you are going to do even simple thinks like battery and memory card changes, and whether you will have access to power for recharging. It will only take one drop of water in an open camera when swapping lenses and you are dead photographically speaking

It's not just the rain that you're competing with, but also the humidity. In this situation, even a waterproof compact or weathersealed SLR will find that tough going, your D40X will most likely die within hours.

I would go with the recommendation of others, in getting a waterproof digital compact, like the Canon Powershot D20, or Panasonic TS-2/FT-2. As well as bringing a compact camera, I personally would bring a simple mechanical film camera, with a cloth shutter and a selenium cell meter. Something like a Nikon F, or a Petri Penta, or an early Praktika. The reason is that none of these things are battery powered which makes them far more resilient to the damp than anything electronic. You could also consider perhaps a Mamiya C33/C330, though these are very heavy.