It's neither a subject nor an object. When I was in school this was called a "noun of direct address". When I took Latin it was called the "vocative case". That is when you use a word to identify the person you are speaking to. The most common use is sentences like, "Bob, come here!", where you may need to identify whom you are addressing, e.g. in a crowded room. It is also used in cases where there is no ambiguity as, basically, just a polite insertion. People sometimes intersperse the name of the person they are talking to in a long string of statements. Like, "And then I went to the store. And you know, Bob, I think that ..."
Usually it is set off with a comma. Like if you say, "I forgot, Sally", you are telling Sally that you forgot some unspecified thing. But if you say "I forgot Sally", then you are telling an unspecified person that you forgot Sally. So we usually write "Good Morning, Jane" rather than "Good Morning Jane". That's more of a greeting than a salutation, i.e. it's a complete sentence. "Dear Jane" does not call for a comma because there is nothing to set the noun off from: "dear" is an adjective modifying "Jane", so both together make up a noun (phrase) of direct address.
I suppose you could add words to the sentence to turn it into the object of a verb, like say that "Good Morning, Jane" is an abbreviated version of "I hope that Jane has a good morning." But I think that's a little strained. You're completely restructuring the sentence.
(I see RegDwight uses the term "vocative" to refer to this usage in English. I'd never heard that before, but whatever, it's the same idea.)