Can I leave indoor wooden furniture unfinished or will it grow mold?

I purchased some furniture that has wooden elements to it that ended up being unfinished. Should I be concerned about this?

No.

It's extremely common for certain parts of furniture, even fine furniture, to have no finish on them. Drawer boxes, the inside surfaces of cabinets and chests of drawers, the backs of pieces intended to go against a wall, seat supports in chairs and the undersides of tabletops have all been — and continue to be — left as bare wood. If you check the relevant parts of furniture that you already owned I'm sure you'll discover examples, in solid wood as well as in plywood.

Leaving wood unfinished can actually be a perfectly acceptable option even in high-humidity parts of the house like kitchens and bathrooms.

During the more humid months, will these elements grow mold on them?

While mould spores are omnipresent in most environments bare wood isn't a great food supply for many fungi, and indoors in a typical domestic interior* the conditions aren't favourable for the growth of moulds in general and certainly not those that can digest wood. One good way for the non-woodworker (and some woodworkers!) to look at this is that bare wood is no more prone to mould growth than the exposed edges of paper in books, and nobody gives them a second thought.

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*Certain levels of cleaning/dusting, heating during the colder months and good airflow being presumed.