Is there a word analogous to 'shooting yourself in the foot'?

The term backfire means

(Of a plan or action) rebound adversely on the originator; have the opposite effect to what was intended: overzealous publicity backfired on her

[Oxford Dictionary Online]

Similarly boomerang

(Of a plan or action) return to the originator, often with negative consequences: misleading consumers about quality will eventually boomerang on a carmaker

[Oxford Dictionary Online]

And blowback

(chiefly US) The unintended adverse results of a political action or situation: this is the blowback from all those aggressive public health campaigns

[Oxford Dictionary Online]

I'm wondering if there's a word that describes the phenomenon where some party takes some action to remedy a situation and the result of that action makes the original situation worse.

First, "shooting oneself in the foot" doesn't necessarily imply that you were trying to remedy a bad situation, only that your actions have caused yourself misfortune in some way.

However, the examples you cite both sound more like the Streisand effect, in which attempting to suppress or censor something ironically results in an increase in its general popularity.

If you were looking for a more general word, there are a lots of words which indicate different degrees of this sort of mistake or error, like gaffe, fumble, and backfire. Gaffe and fumble especially imply that you've caused yourself misfortune in some way, though they tend to apply to verbal actions, as in,

The politician committed this election cycle's biggest gaffe when he gave reporters a piece of his mind, without realizing their cameras were rolling.

Backfire implies that you've caused yourself misfortune in a more general way, as in,

The plan to donate the company's excess revenue to charity backfired when it was revealed that the charity in question was the CEO's foundation.

Counterproductive: thwarting the achievement of an intended goal; tending to defeat one's purpose.

This is basically the adjective form of "backfired".

Backfired: to fail unexpectedly; to fail with an undesired result.

Not a noun, but a verb. A similar word is to have an action boomerang: to recoil or return unexpectedly, causing harm to its originator; backfire.

Shooting oneself in the foot is erroneously thought to have shifted meaning from the intentional act in WWI of soldiers to escape battle (a somewhat befeficial outcome) to the now popular meaning of to do or say something stupid which causes problems for you. (TFD) However, this appears to be a misconception.

An early example can be found in

a sad report in the Appleton Crescent newspaper of August 1857: “Mr. Darriel S. Leo, Consul to Basle, accidentally shot himself through the foot, four or five days ago, in a pistol gallery at Washington, and died on Sunday of lockjaw.” (WorldWideWords)

The author concludes

I’m sure the expression shoot oneself in the foot derives from such accidents, usually the result of incompetence, and has led to our current meaning of making an embarrassing error of judgement or inadvertently making one’s own situation worse. That men did it deliberately as a way to avoid combat is only a side meaning.