Learn Polish: Zarząd bierze odpowiedzialność za ten błąd. - The board of directors takes responsibility for this error.

In Polish, when za means 'for' in the sense of a reason, blame, or exchange (like taking responsibility for something, apologizing, or paying), it is followed by the Accusative case. If za meant 'behind' a physical object, it would take the Instrumental case instead.

The verb bierze needs a direct object in the Accusative case. Why doesn't odpowiedzialność change its ending?

Odpowiedzialność is a feminine noun, but because it ends in a consonant (specifically the common suffix -ość), its Accusative form is exactly the same as its Nominative form. You don't add an -ę like you would for feminine nouns that end in -a.

Is ten błąd also in the Accusative case here?

Yes, it is! Błąd is a masculine inanimate noun (a thing or concept, not a person or animal). For masculine inanimate nouns, the Accusative case always looks identical to the Nominative case. The pronoun ten also stays exactly the same.

Bierze means 'takes', but what is the dictionary form (infinitive) of this verb?

The infinitive is brać. It is an irregular verb, so the stem changes noticeably in the present tense: ja biorę, ty bierzesz, on/ona/ono bierze, and so on.