I've seen this happen quite a lot in arguments about sensitive topics, such as racism, sexism or homophobia.
When presenting a potentially controversial stance, someone might cushion the blow by saying something along the lines of:
"I am not racist, but..." (insert controversial statement here)
I don't have an issue with this, I'm interested in replies such as:
"You say you are not racist BUT... And that says a lot"
Now, this strikes me as a fallacy, but I'm not sure which one it is. The controversial statement in the first example might very well not be racist / homophobic / sexist / etc. at all once you examine it. Using that particular pattern doesn't change a thing about the statement, it might simply reflect that the speaker is aware that it is a controversial one and is wary about how it can be received by the audience.
E.g.: a certain ethnicity is overrepresented in the criminal population
"I'm not racist, but statistics show that race X is three times more likely to commit crime A"
"'I'm not racist, but...' you clearly are"
There are a number of valid objections that can be raised against the first statement, but such a reply attempts to defuse the sentence (and discredit the speaker) by simply appealing to form, igoring the content. And this tactic is often being used to silence someone without replying in the merit of their argument.