Yup. The corporation MUST stay valid while it holds any NFA weapons. If you don't pay your franchise taxes or do something to otherwise dissolve the corporation, you (as an individual) would be illegally in possession of an NFA weapon. That is, the corporation would be considered 'dead' and you (as an individual) would be holding an NFA weapon not registered to you.
You can sell it prior to dissolving the corporation, or if you can get a signoff, you (as an individual) can even transfer it to yourself, but once the corp dissolves, all NFA weapons in its possession must be gone (either by transfer or by destruction).
IMHO, ownership as an individual is always preferable to onwership as a corp. Once you own it as an individual you have no duties (reporting, franchise taxes, etc) until you decide to sell it. Although the corp route gets around the fingerprinting and CLEO signoff, I think its much more hassle in the long run. But I also know its the only option for many people.