Yes, you can run HDMI out to DVI in. It is done quite commonly these days. The Denon 3910 reportedly has the HDMI/DVI Blacker than Black bug when doing that, however -- NOTE: firmware fix on the way.
That bug aside, HDMI is, by design, backward compatible with DVI. An HDMI/DVI adapter cable or adapter plug literally just switches wires around to the right pins for the different style plug -- there is no active processing going on in the cable or adapter plug. When the source and destination connect, a handshake is performed that tells the HDMI source the display wants to be thought of as DVI. The source then locks into the DVI standard. [HDCP -- copy protection -- compliance is checked at the same time.] The same happens in reverse if you hook a DVI source to an HDMI display.
Keep in mind that DVI does *NOT* include audio signals. So if you want audio to get there as well you will need to run separate audio from the source to the display.
If both source and destination are discovered to be HDMI during this handshake, then there are ALTERNATE standards available which provide higher bit depth and YCbCr format vs. RGB format for the video. The handshake then goes on to select the best bandwidth/format supported at both ends. This is for future expansion since commonly available displays today that include HDMI input don't yet take advantage of any of this -- they use only the DVI standards albeit via the HDMI plug.
The HDMI transmitter and receiver circuits mandated by the HDMI standard have additional electronic equalization which allows for error free connections over longer cables than with DVI to DVI. Presumably that also makes the HDMI circuits a bit more expensive. There is no intended difference in the cables however. You just have to get the right plug at each end.
As far as I can tell, it is *NOT* safe to assume you'll get any advantage of this if you cable HDMI out to DVI in or vice versa. As usual with cabling limits, the specific characteristics of the chips in your particular source and display, cable construction, and sources of interference will all play a role in how long the cable can be before you start to see problems. There may easily be variations within different manufacturing runs of a given model from any vendor, although all will meet a minimum level of compliance with the standard.
As I understand it, DVI cable lengths up to only roughly 10 feet are guaranteed to be safe, but many people have successfully used longer cables, and cable companies will cheerfully sell you longer cables. That is, there's enough safety factor in the standard and in how manufacturers have implemented it to give you a pretty good shot that longer cables will work.
Companies like Gefen sell DVI repeater boxes which pass through a regenerated signal for folks having problems. Make sure you get a box that's intended for Home Theater use (i.e., it passes through HDCP compliance from the display) as opposed to boxes designed for connecting computers to their displays. The Gefen and Dtronics DVI switcher boxes also regenerate the signal. Thus you can use a maximum length cable on either side of them. That is, they act as a repeater as well as a switcher.
A number of Forum Sponsors here sell HDMI/DVI cables and adapter plugs. You might want to check the Power Buys forum for any deals currently available.
I believe even Monster Cable offers such cables through places like Best Buy.
Be aware that the markup on cables is enormous. Before shopping, be sure to learn the Home Theater enthusiasts pledge: Open your wallet and repeat after me, "Help Yourself!"