Then it is better to use "producera"; "Filmproducenten producerar film" = "The movie producer produces movies".
(Actually I personally would rather translate "tillverkar" to "manufactures", and "producerar" to "produces". Though they overlap a lot, and dictionaries translate both to both.)
In the context of mass producing, it would be tillverkar. However, for producing only a few items of furniture (e.g. in a small mom-and-pop workshop), it would be bygger, or possibly snickrar if made by an hobbyist. And if we're talking about putting some furniture together (i.e. after going to Ikea), it would be sätter ihop ("assemble").
I hope that covered most scenarios. :)