Can it be shortened to 'I live at the Burj Khalifa in Dubai'?
You should always remember to use the past participle in passive constructions like the one above.
In Table 5-7 Prepositions of Place on page 138 of Azar & Hagen's Basic English Grammar, we can find a very good summary of how the different prepositions of place should be used mainly when speaking about people's locations (I have extracted the ones of interest to answer your question):
(b) Ned lives in Miami.
in the United States.
in North America.
A person lives in a city, a state, a country, a continent.
(c) Meg lives on Hill Street.
A person lives on a street, avenue, road, etc.
(d) She lives at 4472 Hill Street.
A person lives at a street address.
(f) Ivan is at work.
In (f): Ivan is working at his office (or other places of work).
Then, on page 139 there is Exercise 25, where students are expected to complete the sentences with in, on, or at. Sentence (4) is to be completed as follows:
2. He lives at 5541 Lake Street in Toronto, Canada.
Although it wouldn't be incorrect to say "in Toronto, in Canada," the second "in" would sound redundant because everyone knows Toronto is in Canada.
Therefore, the omission of the second preposition introducing the bigger place very much depends on how well-known it is that the smaller place (a district, a village, a town, a city) is situated within the larger one (a city, a state, a country).
For workplaces, "at" is preferred as indicated by the explanation of (f) above. When we speak of workplaces, we refer to the place as an address rather than as a physical place inside which you are when you work—that's why at is preferred.
That said, I'd correct your sentences as follows:
1- I live in Manhattan, New York. (Everyone knows Manhattan is in New York.)
2- I live at the Burj Khalifa in Dubai. (Burj Khalifa is a building in Dubai.)
3- I work at the Victoria Tower in Delhi. (Since Victoria Tower is a building, not a district, "in Delhi" is required.)
4- I live at Shivaji Nagar in Serampore.
4'- I live in Shivaji Nagar, Serampore.
I think the choice between (4) and (4') depends on how Shivaji Nagar is conceived (whether as an urban development or as a district) and on how familiar potential interlocutors are with Shivaji Nagar belonging to Serampore.
As an aside, when optional the inclusion of the second preposition may be more usual in a conversation. In writing, it will tend to be omitted.