Do shows like "House of Cards" need political clearance before their release?

I'm going to answer for the United States. No. Private media does not require any clearance by the Public/Government to produce, distribute, air, or sell any work of art, critical of the government or not. It's protected by the Constitutional guarantee of Freedom of Speech and Freedom of Press, and even given special protection via copyright law.

This may differ for other countries. House of Cards was originally a British Broadcasting Corporation (BBC) miniseries. The BBC is the public service broadcaster of the United Kingdom, a government organization. Whether the BBC needs permission from the UK government would be controlled by UK (and maybe European Union) law. Evidently, it's charter from the government outlines it's independence from purely political control, but the charter is regularly changed, and the government has changed who controls the BBC from the Board of Governors to the BBC Trust as recently as 2007. It's not as simple as the Private sector media rights, and considering the BBC is still the biggest media source in the UK...

Some countries do have strict rules on what can be aired. In 2011, The Chinese Government banned movies that featured Time Travel of all things:

Usually the protagonist is from the modern time and for some reasons and via some means, travels through time and all the way back to the ancient China where he/she will constantly experience the “culture shock” but gradually get used to it and eventually develop a romance in that era. Though obviously the Chinese audience is fond of this genre of shows, the country’s authority … is not happy about this trend and calls a halt to the making of this type of drama.

[Chinese Journalist Olivia] writes that the State Administration of Radio, Film, and Television announced that “the producers and writers are treating the serious history in a frivolous way, which should by no means be encouraged anymore.”

Emphasis mine. Other examples from "moderate" censorship (Thailand) to complete and extreme censorship (North Korea).

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That said, it does happen, by intimidation. The McCarthy era of 1950's Red Scare made many people in Hollywood afraid to produce anything that may get them accused of being a Communist Spy:

The Hollywood blacklist—as the broader entertainment industry blacklist is generally known—was the mid-20th-century practice of denying employment to screenwriters, actors, directors, musicians, and other U.S. entertainment professionals because of their suspected Communist sympathy or membership in the Communist Party. Artists were barred from work on the basis of their alleged membership in or sympathy with the Communist Party USA and refusal to assist investigations into the party's activities. Even during the period of its strictest enforcement, the late 1940s through the late 1950s, the blacklist was rarely made explicit or verifiable, but it directly damaged the careers of scores of individuals working in the film industry.

The first systematic Hollywood blacklist was instituted on November 25, 1947, the day after ten writers and directors were cited for contempt of Congress for refusing to testify to the House Committee on Un-American Activities.

Defacto censorship and "clearance" requirements, just not by law, but by threat of force.

There is also a slight case for Prior Restraint in cases of National Security, but this would never apply to political satire like the House of Cards.