Learn Russian: Машина весит полторы тонны.

The word for "one and a half" changes based on the gender of the noun. Because тонна (ton) is a feminine noun, we must use the feminine form полторы. For masculine and neuter nouns, you would use полтора.

Why is тонны in the genitive singular? Shouldn't it be plural since 1.5 is more than one?

In Russian, the fractions полтора and полторы act grammatically like the numbers 2, 3, and 4. When the number phrase is in the nominative or accusative case, the noun that follows must take the genitive singular form. So, the dictionary form тонна becomes тонны.

What case is the phrase полторы тонны in this sentence?

It is in the accusative case. Verbs of measurement, like весить (to weigh) and стоить (to cost), use the accusative case to express quantity (often called the "accusative of measure"). Because the phrase is inanimate, its accusative form looks exactly the same as its nominative form.

Could I use the verb весить to say "The mechanic is weighing the car"?

No, весить specifically means "to have a weight of" (a state). If you want to talk about the physical action of measuring how heavy something is, you need a different verb: взвешивать.