What I'd heard about the baby-cut carrots was a wheat-based enzyme used to keep them nice and bright orange (not a coloring per se), which posed a problem for Passover, but not the rest of the year.
There are the problems with fruit grown in the first 3/4 years, as well as all sorts of special rules for produce grown in Israel.
Also, there's some discussion about the direct product of a forbidden hybridization, especially if it was done by Jews; but this tends not to be an issue today.
Some produce can occasionally be coated with a shellac-based glaze, which can be derived from secretions of the lac bug. The late Rabbi Abraham Blumenkranz treated this as a serious year-round issue in his yearly Passover book; the question also comes up with regards to some confectioner's glazes (such as may be found on chocolate-covered raisins). Shulchan Aruch writes "secretions of bugs, other than bee-honey, are permitted; though some prohibit." Some of the commentaries wondered why this would ever be practical, but here we are today. In fact, Rabbi Moshe Feinstein permitted the glaze, while Rabbi Aharon Soloveichik prohibited it (and would not allow the dairy diner in Washington Heights under his supervision to sell chocolate-covered raisins).