The Gemara (Shabbos 108a) discusses whether we can write tefillin on the skin of a kosher fish. The Gemara answers that we will have to ask Eliyahu HaNavi:
אי פסקא זוהמא מיניה אי לא פסקא זוהמא מיניה
The Chidushei HaRan writes that at the time of Eve's sin, the snake transmitted zuhama (some sort of filth) onto her and the rest of Creation. When the Torah was given at Mount Sinai, the filth was removed from all categories of creatures in the world that had representatives present at Mount Sinai. This excludes fish, who could not attend. (Since their absence was not under their control, however, perhaps the zuhama was removed from fish, too).
The Sefas Emes (ibid.) writes that the zuhama only leaves animals entirely when they are ritually slaughtered. The Sefas Emes suggests that fish have zuhama because there's no way to remove it from them (shechita is not applicable). Perhaps the Sefas Emes is extending the Ran's idea (who he quotes), and means that there is no shechita for fish because it won't accomplish anything. Animals, however, that were at Sinai, can have their zuhama removed through ritual slaughter.
(As far as birds, the Sfas Emes points out that, even according to the opinion that birds only require nechiras simanim mid'oraysa, that would also function to remove the zuhama).
Additionally, the following perspective comes from Breslov.com:
Since fish live under the water and are thus concealed from view, they are protected from the gaze of the seventy nations of the world. As mentioned above, kosher animals require shechita in order to remove the negative spiritual energies of immorality they have absorbed. Therefore, the righteous, who by definition are individuals who are very far removed from immoral behavior are compared to fish, who are also very far removed from the spiritual energies of immorality. This is why Yosef (Joseph) was blessed that his offspring should be like fish, as the verse says, "May they [Yosef's offspring] proliferate abundantly like fish within the land." (Genesis 48:16) Yosef is the only Biblical figure to be referred to as "HaTzaddik" (the saintly), because he resisted temptation and refused to have any immoral association with his master's wife. Just as the negative spiritual energies emitted from the eye have no negative effect upon fish in the water, similarly the offspring of Yosef are protected from the evil eye and the spiritual energies of immorality associated with the eye.
An additional reason fish do not require shechita is that they live in the water. Besides the fact that water acts as a protective shield against the negative spiritual energies of immorality emitted from the eyes of the nations of the world, it also contains the spiritual energies of purity associated with the hereafter and the Messianic era, when morality and the knowledge of Hashem will prevail, as the verse says, "For the earth [in the Messianic age] shall be full of the knowledge of Hashem, as the waters cover the sea." (Isaiah 11:9)
Fish were protected from the spiritual energies of immorality generated by the generation of the flood because they were concealed by the waters. Therefore, the fish didn't need the spiritual cleansing of the ark that the surviving animals and birds required, as the verse says, "And all flesh that moves upon the earth expired [in the flood]--among the birds, the animals, the beasts, and all the creeping things that creep upon the earth, and all mankind. All in whose nostrils was the breath of the spirit of life, of everything that was on dry land died [The verse mentions only the death of land creatures, implying that Hashem spared the fish, because they did not participate in man's sins (Talmud: Zevachim 113b)]." (Genesis 7:21, 22) And thus, there is no need to perform shechita on fish.
From a standpoint of practicality, also, just taking fish out of the water kills the fish, though not instantaneously. Even if possible, it is not practical to require shechita of fish.
