Hi, The Tampa, Florida man won the state lottery. (No comma after 'Florida'?) The July 15, 2015 meeting has been canceled. (No comma after '2015'?) February 1, 1956 is my date of birth. (No comma after '1956'?) Gary, Indiana, is my native heath. (But, a comma goes after 'Indiana'; if there isn't a comma after 'Indiana', it looks as though we're using a vocative by telling a guy named Gary that Indiana is my native heath.) Do you agree with my punctuation, or lack thereof, in the foregoing examples? Thank you.
The Tampa, Florida man won the state lottery. (No comma after 'Florida'?) Correct. Though, I'd write that as, the man from Tampa, Florida ... The July 15, 2015 meeting has been canceled. (No comma after '2015'?) Correct. But again, this could be written as, The meeting on July 15, 2015 has been cancelled. (I believe the UK and US use different date formats so you should check into that as well) Third one seems fine as well. Gary, Indiana, is my native heath. I don't really understand what you're trying to convey with this sentence, so I can't really comment on it, but if the sentence can be misinterpreted without the comma then leave it in.
"Modern" usage, your lack of commas is correct. "Traditional" says incorrect. Personally, I go with the traditional approach.
Mad Regent, You'd leave out the comma after 'Florida' as well in this one, correct? Or does its omission look as though the state of Florida won the lottery? Does momentary ambiguity arise with the absence of a comma after 'Florida'? The man from Tampa, Florida won the lottery.
Reads fine to me. I understand it OK. I can't comment on others, of course, but you need to have some faith in your readers ability to understand. Also, the comma defines the separation between locations, not the actual sentence itself, so it's still considered as one clause. But as @shadowwalker said, there are traditional methods to it. Commas are always under debate in some form or another. Sometimes, if I'm unsure about comma usage myself, I just look around the net for similar sentences and see what other people do, and try to get an overall consensus on what's commonly used.