Who's going to take care of the dog tomorrow?

Literal

Tomorrow [topic-は] who [subject-が] dog [possessive-の] care [object-を] do [question-んですか]?

犬の世話をする — 'to take care of the dog.' 世話 ('care, looking after') is a common noun. 誰が asks 'who' as the subject. ~んですか adds explanatory tone to the question.