Learn Finnish: Utopiassa eletään ihanteen mukaan. - In a utopia, one lives according to an ideal.

In Finnish, the passive isn't just for actions done to an object. It's frequently used with intransitive verbs (verbs without an object) to express a generic, collective action. Here, eletään simply means "people live" or "one lives" in a general sense, without specifying exactly who is doing the living.

Could I say Utopiassa ihmiset elävät ihanteen mukaan instead?

You could, and it is grammatically correct. However, using the passive eletään is much more natural and idiomatic in Finnish when describing the general conditions or habits of an unspecified group of people, much like using the generic "one" or "you" in English.

Why is ihanne (ideal) in the genitive case (ihanteen)?

The word mukaan (according to) is a postposition. The vast majority of Finnish postpositions require the word immediately preceding them to be in the genitive case.

Why does the -nn- in ihanne change to -nt- in ihanteen?

Words ending in -e undergo reverse consonant gradation, meaning they change from a weak grade in their dictionary form to a strong grade when inflected. The weak -nn- strengthens to -nt-, and the standard genitive ending -en is added to the stem, giving us ihanteen.