Provided that your graphics card is able to run at the native resolution of the HD TV - usually either 1280x720, 1280x768 or 1920x1080 - a HD file will display at its true resolution (i.e. 720p or 1080p).
Also, like a standard def DVD can be upscaled to your monitor's resolution, a 720p file can be upscaled to 1080p. Just maximise the playback software window.
@Karym - DVI itself is perfectly capable of carrying a HDCP signal, it is dependent on the equipment at each end to implement/interperate it. However, HDCP should have been stripped out from downloaded BRD/HD-DVD sources. Even if they're not, its been proven that its not difficult to do so, e.g. Slysoft's AnyDVD can do this on the fly.
But I have found that VGA can give a washed out look to images, e.g. the red lightning bolt on the death proof poster appears pink on my HTPC.
@Curly - with the right codecs (e.g. CoreAVC as mentioned by Munkey) and with the media centre PC set to the TV's antive resolution it will play downloaded HD content fine. Whether thats a DivX 6.7 file or a ripped Blu-Ray or HD-DVD movie.
The optical drive however won't be able to read blu-ray or HD-DVD discs. As an upgrade route there are slot load HD optical drives out there but for the price I'd wait until for a true victor in the high def war...