Learn Finnish: Onko tuo muovinen tuoli kallis? - Is that plastic chair expensive?

It comes from the noun muovi (plastic). By adding the suffix -nen, you turn a material noun into an adjective that means "made of that material". So muovi becomes the adjective muovinen.

Why can't I just say tuo muovi tuoli like we do in English?

In English, we often use nouns as adjectives directly, like "plastic chair" or "glass table". In Finnish, you generally must use the -nen suffix to change the material noun into a proper adjective first before it can modify another noun.

How would the word muovinen change if the chair was in a different case?

Like all words ending in -nen, the basic stem changes to -se- before adding most case endings. For example, if you said "I bought that plastic chair", the words would take the genitive ending: Ostin tuon muovisen tuolin. If they were in the partitive case, the ending becomes -sta, making it muovista.

Does the demonstrative pronoun tuo also change if the case changes?

Yes, demonstrative pronouns like tuo (that) must also agree with the noun they point to. Here, tuo, muovinen, and tuoli are all in the basic nominative case. If you were sitting on that chair, they would all take the adessive ending: tuolla muovisella tuolilla.