"Where do you put up?" is used to mean where are you staying temporarily, but it is more often used with an object:
A: Where do you put up visiting relatives?
B: We have a guest room
A brief tour with the google reveals this advice from "15 Errors most Indians make while speaking English Vinglish":
Coming back to the place of residence -- I often hear a common usage
when it is asked that "Where do you put up?" -- This sentence is
completely and utterly incorrect. This error has passed on as a
"sophisticated" way of asking "Where do you stay?". I don't know who
started this, but it is incorrect and should not be used.
This seems a bit harsh. "Where do you put up has a long history in the language, although it's a bit dated without its object. From a conversation found via the Ngram viewer in the 1843 book Brother Jonathon:
"[W]here do you put up?”
I told him.
“At the Crown? That's odd. Why, I put up there. Well, I'll look in upon you, and hear how you have succeeded."
The Ngram viewer reports no instances of "Where are you put up?"
"Where are you putting up?" seems a valid transposition to the progressive, but it likely clashes with the idiomatic "putting up with something" meaning to endure that thing.