Can the Earth's magnetic field just "dwindle away," as a forthcoming movie "The Core" proposes? And if this were somehow to happen--what dire consequences (also described in the film) could we expect?

Sleep soundly: the premise is far fetched, and even if it did
somehow happen, you would probably not notice any difference.

Could it happen? The magnetization of the ocean floor (among others)
shows that the north-south "main magnetic field" of the Earth does
occasionally reverse its north-south polarity, at irregular intervals
averaging about half a million years. Right now the intensity of this
field is decreasing at about 7% per century. That is a typical rate of
variation, suggesting that if a reversal happens, we will have ample
notice.

However, at the reversal itself the field is not necessarily zero.
Other, more complex parts of the field exist, and right now it looks as
if these are soaking up magnetic energy lost from the main north-south
field. At the time of reversal, the Earth may have for a while 4, 6 or more magnetic poles, and a somewhat weaker field--but it is not expected to lose its magnetism altogether.

And yet, if it ever did... would we be exposed to radiation bursts
from the Sun--the kind which erupt of the order of once a year (or less)and can imperil future astronauts on their way to Mars? No, because we
are shielded by the atmosphere, an absorber comparable to 10 feet of
concrete. Solar bursts cannot penetrate that thickness. It is true thatthe magnetic field of Earth deflects the fast protons of those bursts
even before they reach the top of the atmosphere--but that magnetic
shield fails near the magnetic poles, yet no extra radiation is detectedthere at ground level.

Without magnetic protection, the solar wind emitted by the Sun would
also reach the atmosphere. Could it perhaps strip our atmosphere away?
Maybe, given a few billion years, but not quickly. Venus lacks a
magnetic field and experiences a stronger solar wind, being closer to
the Sun, yet retains a very dense atmosphere. Mars, without a
global-size magnetic field, has only a thin one--but the gravity holding down its atmosphere is only 1/3 of ours.

So sleep soundly, even if have watched the movie.

Dr. David Stern is a retired NASA physicist. In "The Great Magnet, the
Earth," a set of about 20 linked web pages, he tells the story of
geomagnetism, starting with William Gilbert, physician to Queen
Elizabeth I, who in 1600 explained the magnetic compass. The site also
covers magnetic reversals and its home page is on
http://www.phy6.org/earthmag/demagint.htm or
http://pwg.gsfc.nasa.gov/earthmag/demagint.htm