My statistics teacher repeated the statement "Under torture, the data may yield false confessions" several times, but without giving a concrete example of what it means. Can you give some good examples that show the aim of the statement?
What is meant by "torture" is ambiguous. However, I believe that subjecting data to procedures for which it is not intended is a form of this torture. Anscombe's quartet is a classic example of this, subjecting four sets of data to linear regression when three of them clearly do not fit the assumptions.
A second kind of data torture is overfitting, defined many places including this one.
This is known as "data dredging". You can read wikipedia.
The simplest case is when testing 1000 independent hypothesis each with type I risk 1%, you can be almost use one of them will be positive, as a matter of chance.
I prefer the term "cherry picking", which may result from testing tons of hypotheses using unadjusted alpha-level (same as @Benoit Sanchez's).