> ...okay, actually, after saying it a few times, it doesn't sound quite as awful. But what about "Get that dog out of here!" Isn't "out" serving the same purpose as "off" in the previous example? I *know* "Get that dog out here!" isn't correct, so you do know what the difference is?

Basically, no. I did some Googling and came up with the link below, which says, "Do not use extra prepositions when the meaning is clear without them," and goes on to list a bunch of examples, some of which seem too trifling to me -- odd, considering that this particular page is permissive about ending sentences with prepositions, which is a more traditional grammatical sticking point.

Still, I remain in agreement about the principle being expressed: prepositions are mechanical words, function to verbs' and adjectives' form, and so it makes sense if the rules concerning their use are stricter and involve minimizing their use, whereas greater license makes sense with other parts of speech that contribute more to the flavor of the text. For me, "off of" is redundant and horribly awkward.

As a side note, I ran across something that suggests "off of" is primarily American slang ("Get offa me!") and not particularly used in other English-speaking areas. Probably the difficulty of finding a more solid reason why it's grammatically incorrect is why it's gradually been showing up in more formal writing.