Consider this example:
There is some milk in the fridge.
What would be the question to which you could offer that sentence as an answer?
Do we have any milk in the fridge?
Where is the milk?
The first one - so, there is is used to say that some milk exists in the fridge, and therefore it is a dummy existential subject. As professor Lawler explained below, it doesn't belong to any particular word class or part of speech.
If you ask: Where is the milk? Or Where in the fridge is the milk? I'm looking, but I can't find it. Your replies would be:
The milk is in the fridge.
The milk is on the top shelf; it's right there in front of your nose.
Since we are answering where the milk is - there is an adverb of place. When we are explaining where something is, the thing has to be specified, hence the definite article.
So in your example:
There is the man.
We know which man we are talking about. That one, over there. We are pointing at him, showing his location. So there is an adverb of place.
If you say:
There is a man (in the street).
This would mean that a man exists in the street - there would be a dummy subject in this example.
1) I'm not saying that there is a grammar rule by which the use of articles can determine which part of speech 'there' is. But in these examples articles help by clarifying the intended use.
2) A very interesting article about lexical categorisation by G.K. Pullum was recently brought to my attention on ELL. It explains that the way words are categorised in dictionaries often isn't very accurate. So, even though dictionaries include entries for a pronoun 'there', this categorisation can be questioned.