All picture settings need to be checked.
Yes they may need different grayscale calibrations per HDMI, per device.
If you are asking, say example.. I have 1 HDMI in the TV and switch between my HD DVD player and my DVD (or any other player for that matter).
Yes, there needs to be a setting for each individual player (settings should be saved per device, or as my LCDs, they save per switched device).
Is it really that much of a concern for most people, NO.
Does it make a difference, 100% absolutely yes.
Same question would apply if you had 3 HDMI ins to the TV, each one should be calibrated seperately for each input device .
Another example, since you bring up grayscale in particular.
One of my HD DVD player has a setting called Enhanced Black Level. My DVD and BD (PS3) do not have this setting, which will totally change your grayscale for that particular device (and changing grayscale, WILL change the color. So that will need to be checked, as well as sharpness, etc..)..
EDIT; this applies to EVERY input on a TV (RF, VGA, each HDMI, RGB, etc...), as each signal will have different settings coming in (some may be filtered, enhanced, etc... So yes this applies to HD DVD, BD, DVD, etc.. and yes HDTV)
Hope that clears up some of the confusion.