Mike,
With oil-based polyurethane you have durable solids (the stuff that is left behind leaving the protective finish) that are dissolved in solvents. This is what allows it to be in a liquid state. While these solvents are fairly volatile, they can take a long time to evaporate, also known as "offgassing".
There is nothing you need to do to "offgass". You can put the cage in a warm, ventilated room, but it's a natural process that you have to let happen.
Assume the cage is done offgassing when it no longer smells.
Regarding time frame, anywhere from a week to a few months. I'm not kidding about that. Polyurethane is a bit fussy and even experience woodworkers often don't take enough care to apply each coat thin enough. This can cause a finish that never hardens or that takes a very long time to do so.
It is especially problematic with these stain/poly hybrids. The first time I used the Minwax product on a cheap desk I applied the first coat way too thick because it was the only way to make the color look even. I could smell the poly months later. I have since learned that this is a common problem with these products.
The good news is that it's a durable finish. That desk held up for years and it was not until a basement flood that I had to throw it away. The exposed wood was trashed but the poly was still in great shape.
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Current snakes:
1.1 Gonyosoma oxycephala - (Silver/Yellow)
1.2 Gonyosoma oxycephala - (Green)
7.5 Gonyosoma janseni - (Black)
0.1 Gonyosoma janseni - (Black & Tan)