Assalaamu alaykum. If a cat is given haram food to eat (like cat food with pork or other haram meat in it), does the cat itself become impure? Does anything that it touches or licks after that become impure? Also, if a human was to eat haram or impure food, would that person himself become impure? May Allaah reward you.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

The cat does not become najis (ritually impure) because of eating najis food, and the same ruling applies to any living being, regardless of whether that be humans or animals. The living being does not become najis merely because of the consumption of an unlawful or najis item, and what it touches does not become najis either. The Maaliki scholar Khaleel, when listing pure entities, said in Al-Mukhtasar, “And the living being, its tears, sweat, saliva, mucus, and eggs, even if it ate najis food.”

An-Nawawi said, “The leftover water remaining in a vessel from which the cat drinks is pure, and it is not disliked to use this water, and the same ruling applies to the leftover water of other animals ... This ruling applies to animals whose flesh is lawful or unlawful to eat. The sweat and leftover water of all animals are pure, and it is not disliked to use it, except for the dog, the pig, and the hybrid offspring of either of them...” [Majmooʻ Al-Fataawa]

Whatever the cat which ate najis food licks or whatever the person who consumed a prohibited substance such as alcohol or pork touches does not become najis unless the cat or person has "visible impurity", like on the mouth of the cat or the hand of the person, or it is known with certainty that the mouth or hand are contaminated with impurity. If that is the case, then whatever the cat licks or the person touches with his hand becomes najis. Minah Al-Jaleel 'ala Mukhtasar Khaleel Al-Maaliki reads:

“If the impurity is proven to be on the mouth or hand of the person who drinks intoxicants by means of observation or a reliable report, then the hand or mouth is declared najis. The same applies to the animals which do not avoid impurities such as cats or the like during the use of water or eating; if the cat licks the water and its color, taste, or smell changed, then it is declared najis. If the water did not change, then it is still disliked to use this water if it is a small amount. As for food, if the cat licks the food, in solid or liquid form, then it becomes najis as well.” [Excerpted with modification]

Shaykhul-Islam Ibn Taymiyyah wrote, “If the cat ate a mouse or the like - of impurities - then after a long time interval its mouth becomes pure through its saliva, given the need. This is the strongest scholarly view in this regard. It was adopted by a group of Imaam Ahmad's students (Hanbalis) and Imaam Abu Haneefah. The same ruling applies to the mouths of children and cattle.” [Al-Fataawa]

In brief, the cat and other living beings do not become najis by merely consuming prohibited or impure food.

It should be noted, though, that scholars held different views as to whether it is permissible or disliked to feed animals impurities. The preponderant view is that it is allowable, as adopted by the Hanbalis and those who agreed with them in this regard. The Hanbali scholar Al-Buhooti wrote, “It is permissible to feed the animal impure substances if it will not be slaughtered or milked soon.”

As for the animals that will be slaughtered soon, in the Hanbali mathhab it is prohibited to feed them impurities so that they do not become jallaalah (filth-eating animals), whose flesh is prohibited to consume in the Hanbali mathhab.

Allaah knows best.