As long as you can still smell it, it's still off-gassing. While this is obvious, its somehow overlooked. Go figure.
That said, there's no way to tell what will be a problem without knowing more about the varnish in question - but at the minimum, don't eat off of anything with an odor. Beyond that, different types of stuff have different abilities to interact with those gasses; some ceramics may hold them briefly, some plastics may co-dissolve them with their plasticizer (plastic is a bad idea in general, it seems), but clear glass will usually only get contamination at the surface, making it relatively easy to clean even to the standard of "take no chances, assume everything is dangerous until proven safe even though nobody trustworthy will ever run those tests on pregnant women deliberately in the first place."
You can actually get clear glass plates that don't easily break, even - just be sure to stay away from anything that appears to really diffract light in a pretty way or that is labeled as "crystal glass" - some of that may have a lead content that can be leached by acidic substances. Odds of running into that in 2011 are close to zero, but I wouldn't say its impossible if you were looking in thrift or novelty stores either..
Keep anything plastic Mom or the child might use out of that cupboard for at least twice as long as it takes it to stop having a noticeable odor - if plastic can leach and absorb plasticizers, it stands to reason it can probably leach and absorb a whole plethora of other hydrophobic molecules too, and without the ability to detect how much, its sort of impossible to say what is a safe amount
I'm not saying twice as long is enough either, but it seems reasonable that the time it takes to drop from "concentrated" to the limits of detection by something as sensitive as the human nose, especially for aromatics, if doubled, could very well be long enough to reduce any exposure risk below what is probably considered unavoidable in the first place.
Freshly opened electronics (big TVs) also have a similar issue with the chemical stink that creates these anxieties, same principles should apply to them, I'd think.
When you're out of the home, pointing a fan at it may help.