Learn Polish: Oni kręcą tu film. - They are shooting a film here.

Polish uses kręcić (to turn or crank) for making movies because early film cameras were operated by manually turning a hand crank. So, kręcić film is a common set phrase that means "to shoot a film".

What case is film in, and why didn't its ending change?

It is in the accusative case because it is the direct object (the thing being shot). However, film is a masculine inanimate noun (a masculine word that isn't a person or animal). For these nouns, the accusative form is identical to the basic dictionary form, so no extra ending is added.

Can I use tutaj instead of tu in this sentence?

Yes! Tu and tutaj both mean "here" and are completely interchangeable. Tu is just a shorter version that is very common in everyday spoken Polish.

Why do we use oni instead of one for "they"?

Oni is the masculine personal pronoun, used for groups of people that include at least one male. If you were talking about an all-female film crew, you would use one instead: One kręcą tu film.