Learn Hardcore Icelandic: Hún býr í húsi.

Because after the preposition í (when it indicates location), Icelandic uses the dative case. Hús is a neuter noun whose dative singular form is húsi.

Why isn’t there an article like “a” or “the” before húsi?

Icelandic doesn’t use separate words for “a” or “the.” Instead it marks definiteness on the noun itself.
• húsi = “in a house” (indefinite)
• húsinu = “in the house” (definite; note the ‑inu ending)

What does the verb býr mean here?

býr is the 3rd-person singular present of búa, which means “to live” or “to reside.” Don’t confuse it with bjóða (“to offer”).

Can you omit the subject Hún and just say Býr í húsi?

Yes, Icelandic verbs are inflected for person and number, so the pronoun can be dropped if context is clear. However, leaving out hún can sound abrupt or poetic.

How would you say “She lived in a house” (past tense)?

Use the past form bjó of búa:
She lived in a house → Hún bjó í húsi.

How do you pronounce hún býr í húsi?

Approximate IPA transcription:
• hún [huːn]
• býr [pʰiːr] (the Icelandic b is unvoiced)
• í [iː]
• húsi [ˈhuːsɪ]

Are there other prepositions for “living somewhere,” or is it always í?

For enclosed spaces (houses, rooms) you use í. Some locations use á (e.g. á Íslandi “in Iceland”). You might also see á hóteli (“in a hotel”) or við in expressions like búa við fjöru (“live by the shore”), but for a house it’s always búa í húsi.

How would you say “He lives in a house”?

Change the pronoun to hann and keep the verb form:
Hann býr í húsi.