Question: Sorry for the catchy title. I want to understand, what should one have to do to disprove the Church-Turing thesis? Somewhere I read it's mathematically impossible to do it! Why?

Turing, Rosser etc used different terms to
differentiate between: "what can be computed" and "what can be
computed by a Turing machine".

Turing's 1939 definition regarding this is:
"We shall use the expression "computable function" to mean a function
calculable by a machine, and we let "effectively calculable" refer to
the intuitive idea without particular identification with any one of
these definitions".

So, the Church-Turing thesis can be stated as follows:
Every effectively calculable function is a computable function.

So again, how will the proof look like if one disproves this conjecture?

Answer: 
Regarding the *Extended* Church-Turing Thesis (meant as "A probabilistic Turing machine can efficiently simulate any physically computable function."):

One possibility is the difference between classical and quantum computers. Specifically the question, "Is there a task that quantum computers can perform that classical computers cannot?" [A recent ECCC report by Scott Aaronson](http://www.eccc.uni-trier.de/report/2010/128/) (see Conjecture 9 on page 5) highlights a conjecture that, if proven, would provide strong evidence against the Extended Church-Turing Thesis.

If one were to disprove the Extended Church-Turing Thesis, it could look like that -- specifically, by demonstrating an efficiently computable task that a (classical) Turing machine cannot efficiently compute.
The Church-Turing thesis has been proved for all practical purposes.

<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.146.5402>

Dershowitz and Gurevich, Bulletin of Symbolic Logic, 2008.

(This reference discusses the history of Church's and Turing's work, and argues for a separation between "Church's Thesis" and "Turing's Thesis" as distinct logical claims, then proves them both, within an intuitive axiomatization of computability.)