While I can't say that I have tried saving an incomplete form in this manner, I will say that the vast majority of pdf forms such as this are designed not to be saved, and many that I have filled out have included a warning that it would not be possible to save the edited portions of them. I thus do not believe that it is possible to save a copy of the form and have it retain its editability.
Some further googling has turned up this PDF stating that while Acrobat can save these forms, Acrobat Reader cannot.
In addition, I found some shareware applications designed for this purpose, but none that I have found support Linux.
Ctrl+S? I think it depends on what version of evince/reader you're using. – Mark Feb 23 '10 at 23:23