Say a group would like to play D&D but without a DM.
Ok.
Can the DM be replaced with a deck of encounter cards, a random encounter table, or some other mechanic for such an experiment?
No, the DM cannot be replaced by anything but a human level intelligence.
You could however still play D&D as a skirmish game using wandering encounter tables. While I wouldn't recommend using 1e as a tactical skirmish game (because it would get really redundent), 1e is well suited to generating random encounters.
I'd say that before 4e, there'd not be much point in that, but 4e would probably play decently as a skirmish game. The monsters are fairly simple and don't require alot of DM tactical mastery to use well, so its less of a problem having the NPC's controled by someone who has a vested interest in seeing the monster 'do badly'. Tactics with monster are usually pretty straight forward compared to 3e, for example, because the monsters usually have one obvious trick and one obvious board position that they are trying to maintain. (Compare with a 3e Noble Salamander or most outsiders (barbed devil for example) with their varied array of possible tactics and long list of powers.) And, 4e can kill alot of time with tactical play and the whole encounter system supports stand alone meaningless encounters alot better than any prior edition.
So yeah, you could do it.
Can a solo game be sustained over the course of leveling up? Throughout a campaign? Thoughts?
I guess that depends on how interesting the players find tactical skirmish games. Alot of people are complaining about how boring and repetitive 4e combat is in the context of a full fledged story line, so while 4e might be my edition choice for a tactical skirmish game I'm not sure that even then it's got enough to maintain interest in combat for combat's sake. However, I remember meeting junior high kids back in the day who had level up characters to high levels simply by picking entries in the MM, rolling up an encounter, and then fighting them, with no real 'DM' required, so I imagine some people would dig it.
The game would still be better with a DM, a map, and at least a thin gruel of a story.