When referring to an entity like a government body, should it be capitalized if referring to is by classification(?).
E.g., if I write:
The City of New York requires us to get a building permit.
Certainly "city" should be capitalized.
However, if I'm communicating in a context where everyone should know what city I'm referring to (the one we are in), when I write:
The city requires us to get a building permit.
Should "city" still be capitalized?
In your second example, "city" should not be capitalized. Words for governmental or administrative units are only capitalized when they are used as part of a proper noun, such as the formal name of a city.
Your first example is correct so long as you're referring to the City of New York, as the formal name for New York. However, if I were simply using the word "city" to disambiguate and not as part of a formal name, I wouldn't capitalize it:
We're only going to the city of New York, not the rest of the state.