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: 
Disproving the Church-Turing thesis seems indeed extremely unlikely and conceptually very hard to imagine. There are various "hypothetical physical worlds" which are in some tension with the Church-Turing thesis (but whether they contradict it is by itself an interesting philosophical question). A paper by Pitowsky "[The Physical Church’s Thesis and Physical Computational Complexity", Iyun 39, 81-99 (1990)](http://edelstein.huji.ac.il/staff/pitowsky/papers/Paper14.pdf) deals with such hypothetical physical worlds. See also the paper by Itamar Pitowsky and Oron Shagrir: "[The Church-Turing Thesis and Hyper Computation](http://edelstein.huji.ac.il/staff/pitowsky/papers/Paper36.pdf)", Minds and Machines 13, 87-101 (2003). Oron Shagrir have written several philosophical papers about the Church-Turing thesis [see his webpage](http://edelstein.huji.ac.il/staff/shagrir/). (See also [this blog post](http://gilkalai.wordpress.com/2010/02/15/itamar-pitowski-probability-in-physics-where-does-it-come-from/).) 

The effective or efficient Church-Turing thesis is an infinitely stronger assertion than the original Church-Turing assertion which asserts that every possible computation can be simulated effciently by a Turing machine. Quantum computers will indeed show that The efficient Church-Turing thesis is invalid (modulo some computational complexity mathematical conjectures, and modulo the "asymptotic interpretation"). I think the efficient Church-Turing conjecture was first formulated in 1985 by Wolfram, the paper is cited in Pitowsky's paper linked above. In fact, you do not even need universal quantum computers to refute the efficient C-T thesis, and it is interesting line of research (that Aaronson among others studies) to propose as simple as possible demonstration of the computational superiority of quantum systems. 

It is also an interesting problem if there are simpler ways to demonstrate the computational superiority of quantum computers in the presence of noise, rather than to have full-fledge quantum fault-tolerance (that allows universal quantum computation). (Scott A. is indeed interested also in this problem.)
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.)