If I may, this needs a bit of explanation. Live plants, if by this one means aquarium plants grown submersed, will not remove any nitrates already in the source water. This is because they prefer ammonia/ammonium as their nitrogen source, and there is usually plenty of this available in an aquarium with fish that are fed. Aquatic plants will only begin taking up nitrate if the ammonia/ammonium is insufficient for their needs (i.e., out of balance with light and other nutrients, and thus the "limiting factor" to plant growth).
Aquatic plants can help with nitrate that occurs from the biological processes in the aquarium. They do this by using ammonia/ammonium and taking it up faster than the nitrifying bacteria/archaea can, which means less nitrite and thus less nitrate as a result. Fast-growing plants such as floating species can really do this, which is why they are often referred to as "ammonia sinks."
Terrestrial plants use nitrate as their preferred source of nitrogen. So terrestrial plants such as houseplants may help if their roots can be submersed permanently in the tank water. Provided the plant is not toxic (many houseplants are), such plants can remove nitrate. I've never had this issue fortunately, so no idea how effective, but others have if memory serves me said it does work.
