Is there a word for when an action has an effect opposite to the one intended?

It's called a Paradoxical Reaction. In general terms, I think you call something like that a paradox.

A common idiom expressing this concept is "to backfire". For example:

• The campaign to reduce smoking backfired.

I just found this term recently and thought that it will be useful for somebody:

The Streisand effect is the phenomenon whereby an attempt to hide or remove a piece of information has the unintended consequence of publicizing the information more widely, usually facilitated by the Internet.

[Wikipedia]

This term is now specific to mass media / Internet, but it will be possibly penetrating in the other relevant fields. However, general concept here is a bit more specific here: forbidding something can possibly rise an interest in something and thus cause a more wide spread (i.e. an opposite effect of intended action).

Counter-productive seems to fit your request pretty closely. The OED defines it as:

Having the opposite of the desired effect, tending to act against the attainment of an objective.

It’s a comparative neologism (apparently originally from US bureaucrat-speak in the 60’s), but is now well-established on both sides of the Atlantic in both formal and informal use:

The drug laws are counter-productive, and David Cameron knows it.
                — Tom Chivers, in the Daily Telegraph

But you’re right, they wouldn’t sue him (even if they had a case) because it’d just be counter-productive.
                — lightlee.tumblr.com [random Google result looking for casual use]

Macmillan Dictionary: do someone​/​something a disservice​/​do a disservice to someone​/​something

to do something that makes people’s opinion of someone or something not as good as it should be To describe her as just a journalist is to do her a disservice. He is doing himself a disservice by allowing his songs to be so badly performed.