Learn German: Ein Teil vom Kurs fehlt heute. - Part of the class is missing today.

It can be both, but the meaning changes! der Teil (masculine), used here, means a fraction or portion of a whole, like a section of a course. das Teil (neuter) refers to a physical piece or component, like a puzzle piece or a spare part for a car.

Why is the verb fehlen in the singular form (fehlt) if 'part of the class' refers to multiple students?

In German, the verb strictly agrees with the grammatical subject. Since the subject is ein Teil (a single part), the verb must be singular (fehlt). English sometimes uses plural verbs for collective nouns, but German usually sticks to the grammatical number.

Could I use the genitive case here instead of vom Kurs?

Yes, you absolutely can! Ein Teil des Kurses is completely correct and sounds a bit more formal or written. In spoken, everyday German, people frequently replace the genitive with von plus the dative case, which combines to form vom.

Does fehlen literally mean 'to be missing' like a lost item?

It does, but it is also the standard German way to say that someone is absent. When students or colleagues are not at school or work, Germans say they are missing (sie fehlen).