Just saw a big debate (in the middle of a completely different debate) about whether it's easy to remove alignment from D&D and whether that will change the game dramatically.
My answers are "yes" and "maybe".
Let's look at how alignment interacts mechanically with the rest of the game.
First, you have the common interpretation of alignment as personality, which is expressed in two game mechanics: a simple bonus or penalty on reaction rolls when negotiating with characters of matched or mismatched alignments, and an xp bonus or penalty for acting in accord or discord with one's declare alignment. The reaction modifier isn't explicit in OD&D, but it shows up in later editions and in the Judges Guild Ready Ref sheets; it's pretty simple to add or remove, or replace with a set of modifiers based solely on real actions instead of abstract alignment. The xp modifier isn't mentioned in OD&D at all, but it becomes an informal rule in some of the basic line; in AD&D, there's a way too fiddly method for tracking alignment change and modifying experience accordingly. Dropping just these rules doesn't affect the game much, really, especially in OD&D.
Second, you have the spell system. Cleric spells that cause harm are actually forbidden for Lawful clerics (or Good clerics, in later editions;) and, some argue, the healing spells are forbidden for Chaotic clerics, or at least this is one possible interpretation in OD&D. As a consequence, OD&D played according to the 3LBBs makes a sharp distinction between Lawful clerics, who can't cause magical harm and who have healing powers, and magic-users, who have several ways to cause damage, sometimes lots of it. Removing this alignment rule in OD&D thus changes the game significantly, making clerics just another spell caster... but if you are using the Greyhawk supplement or AD&D, this sharp distinction goes away, anyways, because of spells like Spiritual Hammer and Flame Strike; dropping alignment restrictions for clerics doesn't cause much change in AD&D.
Conversely, there are a few spells in OD&D, more in AD&D, which specifically mention alignment. There's a good argument that Protection from Evil and Detect Evil in OD&D really have nothing to do with alignment at all, at least under some interpretations. In later editions, especially AD&D, there's a stricter adherence to alignment, so dropping alignment would require removing or re-writing several AD&D spells, perhaps rolling it back to the OD&D interpretations.
Finally, you have magic swords and a few other items, particularly artifacts, which have an intrinsic alignment. This system works about the same across all TSR editions (not sure about WotC editions,) but it's very light-weight, involving only potential damage and possible denial or reduction of the item's powers. This is notable in that it doesn't require the interpretation of alignment as behavior or personality to make it work. Even so, I get the feeling that many GMs keep alignment as personality, but ditch the alignment rules for magic items, thus keeping the more complicated, annoying aspects of alignment.
So, in summary: dropping all aspects of alignment from OD&D won't matter too much, but you lose the interesting optional behaviors of magic weapons and kind of make clerics less distinct as a class. Ditching alignment for a later version, particularly AD&D, makes the game look more like OD&D (and saves a lot of bookkeeping in AD&D.) It only becomes complicated if you decide to rewrite spells linked to alignment, instead of simply dropping them.