Whether the roots will be a problem rather depends what the roots are from; if they are off a tree such as a cherry, you may well find little cherry trees trying to grow off it. In general, though, regular mowing will prevent this problem. However, if the roots are from pernicious weeds such as bindweed, thistle, dock, dandelion and so on, or from plants like bamboo, they will constantly regrow, and faster than the grass during growing season, so those roots need to come out. You presumably know what was growing there before you dug it over?
It is not a good idea to put a layer of plastic beneath any growing area; water will collect because plastic blocks drainage. You could, in theory, lay a geotextile membrane, which is porous, but growth from roots such as bamboo, japanese knotweed or tree saplings/suckers will easily punch through that layer so there's not much point in laying some.
On the subject of weedkiller, which type is important. In many countries, particularly in Europe, pre emergent weedkillers are not available; most other weedkillers such as glyphosate work 'through the green', meaning they kill through the green parts, which should be growing strongly when applied. This latter type of weedkiller will do nothing at all if there is no green growth present and will also not deter any growth of tree roots because they don't kill woody parts of large plants. Other weedkillers such as Paradise will kill off any plants present, but will prevent any and all growth for up to six months - including your grass.
Without knowing what the roots are from, it's hard to say definitively whether you will have a problem after laying grass. If you don't recall much growing there at all, but the area is surrounded by trees or large shrubs, then the roots from those are likely to have been what you found in the soil. Grass should still grow over those, though the presence of many trees might mean the grass does not get enough sunlight, so that's another consideration.
