I understand that sugar serves several different purposes in cooking so it's often not okay to replace with natural sweeteners like stevia. However, when significant amounts of sugar are required, is it alright to substitute a proportion of the sugar required with a natural sweetener?
It depends on the context and the role that sugar is playing in the recipe, and the chemistry or properties of the particular sweetener you anticipate substituting.
In something like a jelly, where the sugar, acid, and pectin must be in the proper ratio in order to thicken the product, no you cannot simply make this substitution.
In most sugar-based candies, including hard candies like lemon drops, lollipops, or candy canes as well as softer sugar based candies like caramels, fondants, nougats, divinities, fudges, and so on, you cannot make the substitution because the sugar is the body of the candy, and its form and texture are critically dependent on the crystallization properties of sugar.
Similarly, many icings are critically dependent on the detailed properties of sugars. Glazes and seven minute frostings would be difficult to substitute. The type of so-called butter cream which is just butter or shortening beaten with powdered sugar might allow a partial substitution, but it would be tricky to get the right texture.
Meringues specifically require hot syrup to help set their texture, so it might work with a syrup made with alternate sugars like honey or maple syrup but not with other substitutes like stevia which do not provide the bulk and form syrups.
In a recipe like lemonade, where the only purpose of the sugar is to sweeten, yes, you can. It will change the mouth-feel and yield, however.
In cakes and other baked goods, you can probably do a portion, but you sugar also performs a role as part of the structure and helps with moisture retention, so you don't want to go overboard in a recipe not specifically designed for alternate sweeteners.