I seem to recall the use of the word “done” should relate only to food preparation (the turkey is done).
Conversely, the word “finished” would be used as follows: My homework is finished.
Anyone ever heard this about “done?”
No.
Sentences like “I have done the dishes,” “This meeting is done,” and “He has done the most for our company,” are all perfectly correct.
Sounds like you had a prescriptivist teacher at some point, who liked making up rules. “Done” is one of the most common words in English. Even used as an adjective, it’s perfectly fine in many senses having nothing to do with food preparation.
Just a few examples from Shakespeare:
That is done, sir. – Merchant of Venice 3.5
That’s all one, our play is done. – Twelfth Night 5.1
All’s done, all’s won. – Henry IV, part 1 5.3
Death approach not, ere my tale be done. – Henry VI part 1 2.5
–Mark
There’s also usages like ‘the done thing’, and it can be used to mean tired (‘done in’) or near death (‘done for’), the latter often being used metaphorically.
You could look at page 740 of the Shorter OED.
This is not really a grammar question; it’s a usage question.
Not only do i agree with the others, i’ve never even heard of this usage demarcation between “done” and “finished” before. The main distinction i would draw between them is that “done” might, in some situations, sound a little more informal than “finished” or “completed.”
Anyway, i’m done here. 
I have heard it. Usually by someone being a bit of a snot. Not even they actually use it that way, but they’re being cheeky.
It’s usually phrased in the form “Cookies [or cakes] get done. People get finished.”
I did find an interesting Motivated Grammar poston it, and it says it well enough that I won’t bother to paraphrase:
It looks to me that the real reason why people started complaining about this usage is that it had two signs of the prescriptivist devil: it was a new usage, and it was a non-standard usage. To be done, the MWDEU reports, supplanted to have done for states of being starting sometime in the 1700s or earlier, which on a prescriptivist timescale somehow counts as “new”. Furthermore, the OED classifies this usage as chiefly Irish, Scottish, American, and dialectal, which to a prescriptivist is just a long way of saying improper. And usually finished sounds fancier than done, which no doubt contributed to the distaste for done.
A high school teacher of mine would say something similar. This would have been about 1990. I think he was dead serious, though.
I don’t think there’s any real distinction, but “I’m done” does sound slightly informal to me.
He really done went and said that, huh?
I had several grade school teachers who had a standard response to a student saying “I’m done.” ----
“How long did you bake?”
You were supposed to say “I’m finished.” instead.
I’m trying to think of what rule-ish thing they were invoking, but it’s all just some silliness like the idiotic less/fewer nonsense.
Heh. I actually thought from the thread title this was going to be about the completive aspect usage of done in some dialects of English.
It’s not a cooking/other activities distinction, I think. “Do”, when not used as an auxiliary, is a transitive verb. It requires a direct object, either explicit or implicit. The student does the homework (or the cooking, or whatever), after which the homework (or the cooking) is done. The student is the actor, not the action, so the student isn’t done; the student is finished, or (better) has finished.
“I’m done” to mean “I have finished” sounds, to my non-American ears, sounds like an Americanism, and a colloquial one. Would it be used in a formal register in US speech?
Really? I grew up in Australia, and have lived for extended periods in the UK, Canada, and the United States, and i’m pretty sure i’ve heard this usage all my life.
I think it might be used a little more frequently in the US and Canada than in Britain and Australia, but i didn’t think any Brits or Aussies would be surprised or confused about it.
Interestingly, though, of the two usage dictionaries i have on my shelf, the British one (Fowler’s Modern English Usage, 3rd edition, by Burchfield) doesn’t have an entry at all for this usage of “done.” The American one (Dictionary of Modern American Usage, 1st edition, by Garner) does have an entry:
For what it’s worth, to my ears “I’m through” in the sense of “I have finished” also sounds American.
I grew up in Ireland, and I’ve been living for the past 15 years in Australia. I don’t hear this sense of “done” very much in Australia, and my Concise Macquarie doesn’t have it. It has “done” in these sense of “executed, completed, finished, settled”, but that’s in the sense of the cooking, etc, being done. For the self-referential “done” it offers “done with”, as in “I’m done with the cooking” or “I’m done with him”. But “I’m done with cooking” doesn’t mean that I have finished a particular cooking project, I think. It would suggest that I’m never going to cook again or, at least, not for a long time.
The Oxford English Dictionary also doesn’t have the “I’m finished” sense of “done”. An action can be done, i.e. completed, but a person who is done has not simply finished an action; rather he is tired, exhausted; beaten, defeated; worn out, used up; incapacitated.
“How long did you bake?” “Since 4:20.” 
I can’t really imagine using any word other than “done” when grilling meat. I guess it’s more informal than “finished”, “completed”, and “accomplished”.
Well, there’s always “ready”.
What we have here is another example of words of Latin or Norman French origin (finished) considered more proper/posh than words of Anglo-Saxon origin (done): which is not exactly a recent development…