Exactly what I was thinking...
Raw wood is a moisture sponge. It soaks it up and then starts to rot.
If your wood cage is not properly treated, you will want to thoroughly dry it out before applying a sealer. If not you will simply trap the moisture in the wood allowing it to rot from the inside out.
There is a huge variety of sealers you can consider. Latex Dry Lock is a personal favorit eof mine. I've used it on Reptile Enclosures, Fish tanks/ponds, as well as it's intended use for sealing concrete.
Dry Lock can be tinted with standard latex paint color additives. I've added up to 6 ounces of coloration and the result was still water tight.
There are also many other sealers that can be used... Standard polyurethane is an option but you will want many coats... Acrylic sealers are an option but they tend to be pricey and I'm not sure they will stand up to a warm bulb shining on them...Rubber sealers should work but can get pricey... You could smear brown silicone on the inside and texture it to look like bark (I have made fake trees like this for a vivarium)...
There are a lot of things that will work... but not sealing the wood will not work for very long...