Your receiver absolutely does not need HDMI. There are few advantages to using HDMI on a receiver, none of which are a big deal:
1) Lets you maybe eliminate a cable or two
2) Lets you hook up multiple HDMI sources to a TV with only one HDMI input
It's almost a necessity to have HDMI on your display, though. Quality upconverting DVD players only upconvert over HDMI. Many cable boxes output a superior picture via HDMI than they do via component. Future high-definition content may only be watchable via HDMI thanks to digital copy protection.
Thankfully, the vast majority of current HDTVs will have HDMI inputs, so you have nothing to worry about in that regard.
Generally, here's the setup:
Upconverting DVD player hooked to HDTV via HDMI cable and to receiver (for audio) via optical or digital coax.
DirecTV HD box hooked to HDTV via HDMI cable and to receiver via optical or digital coax. You may need an inexpensive HDMI switch if your TV only has one HDMI input, or you may use component cables for video instead of HDMI.
You can watch TV and movies and listen to music in surround sound without HDMI. HDMI is primarily a video cable at the current point in time because HDMI is not very prevalent on receivers. In the future, more receivers will have HDMI ports built in, allowing you to eliminate the separate optical and digital coax audio cables, but in the meantime it's not a worry.
Also note: HDCP-enabled DVI is equivalent to HDMI. They are physically compatible. Only difference is that DVI inputs cannot accept audio. However, having an HDCP-enabled DVI input on your TV is really the equivalent to having an HDMI input.
