Do nonpolar crystals form ionic or molecular liquids?

Answer 1

There is no cut and dry answer to this question.

If you are in a low level chemistry or geology class then the typical answer is that ionic liquids only come from ionic solids, some of which are crystals and molecular liquids only come from molecular solids, some of which are crystals.

But non-polar crystals can also be metals. Pure elemental metal crystals are non-polar, but the liquid metal is not molecular. Both the solid and the liquid metals will conduct electricity.

Diamond and Buckminsterfullerenes also form non-polar crystals, but neither is know to exist easily as liquids, ionic or molecular.

Some forms of Silicon dioxide are crystals. When heated hot enough, they do melt. The polarity of the Si-O bond is minor, but melted quartz crystal is a glass, neither an ionic nor molecular liquid.

Answer 2

Molecular. To form an ionic liquid you not only have to be polar, you have to literally be charged--have cations and anions.