Suppose that I have a warm room. Is it possible to condense the heat of the room into a small object so that the room turns cold and the small object very hot?
If it's possible: How? If it's not: Why?
Let me first say that you're using the term "heat" in a colloquial way that does not align with the definition of the term in thermodynamics. In particular, heat refers to the transfer of energy from one object to another, it is not itself a form of energy.
The term that appropriately describes the energy in the room is "internal energy."
Having said this, let's assume that you're referring to the internal energy present in the air because the air has some nonzero temperature. Then we would ask whether we could somehow "condense" the internal energy present in the air so that it is contained in a smaller volume.
Well, you could certainly simply condense the air inside of the room itself to accomplish this using a vacuum pump by sucking up all of the air into a small cubical, thermally insulated box for example.
You could then ask if the air inside of the box would be hotter than the air originally in the room. Well the answer to this depends on how you compressed the air for the following reasons. Let's model the gas in the room as a monatomic ideal gas for simplicity. In this case
If you were to compress the gas adiabatically, meaning with no heat transfer, then $\Delta E = -W$. Since work needs to be done on the gas to compress it, you would also have $W<0$ so that $\Delta E>0$, and as a result, the temperature of the gas would be greater after being condensed (assuming the particle number remained the same).
However, if you were to compress the gas while also extracting heat from it, then you could arrange for the final sample to contain less internal energy and therefore be colder (assuming again particle number remained the same).
There are all sorts of possibilities really, it all depends on what thermodynamic process you perform!
Sure that's possible. Air conditioning is for example something that does this. A fridge does a similar thing but in reverse.