What's the term for denying something so as to raise doubt?

Topical example:

'There was a scuffle outside the local pub. There appears to be no connection with religious extremists.'

...but now you're wondering if there was. It's a nasty tactic used to raise fear and doubt - deliberately calling attention to something by explicitly denying it.

What's the term for this?

In rhetoric, this is called paralipsis or apophasis.

Innuendo, insinuation, overtone & connotation come close.

innuendo: a statement which indirectly suggests that someone has done something immoral, improper, etc

insinuation

: a usually bad or insulting remark that is said in an indirect way

: the act of saying something bad or insulting in an indirect way

overtone : an idea or quality that is suggested without being said directly

connotation: an idea or quality that a word makes you think about in addition to its meaning

The word you're looking for is probably misdirection:

Misdirection is a form of deception in which the attention of the audience is focused on one thing in order to distract it's attention from another.

More broadly, the phenomenon you refer to is a part of Ironic process theory.

A logical fallacy is an error in the logic of an argument that prevents it from being logically valid but does not prevent it from swaying people's minds. http://rationalwiki.org/wiki/Logical_fallacy

Your example looks like a case of media manipulation:

a series of related techniques in which partisans create an image or argument that favours their particular interests.[1] Such tactics may include the use of logical fallacies and propaganda techniques, and often involve the suppression of information or points of view by crowding them out, by inducing other people or groups of people to stop listening to certain arguments, or by simply diverting attention elsewhere. https://en.wikipedia.org/wiki/Media_manipulation