I have heard people mention not having plants grow under trees, I thought because of the loss of nutrients to the tree that might result, but not entirely sure. I was curious to know if letting nasturtiums cover the ground under a lemon tree was a bad idea. I have a bunch of them that are really creeping quickly and are already within the drip line of my lemon tree. They will likely cover the entire area (small as it is) if I let them. Should I prevent this? Pics below..

It is often difficult for plants to thrive under trees because the tree is using all the resources (water and nutrients). Nasturtiums are pretty tough. I'd let them go wherever they want to - that's a very nice combination.

Thanks it would be nice to keep them. I can't let him go the other direction because they would overrun a lot of other plants

Plants under trees . . . some of the issues. Where I live the native Oaks shouldn't be watered, hence don't plant under the Oaks if you plan on watering. The other issue I've run across is that trees create shade, often deep shade and not much will grow in deep shade. I wouldn't worry too much about robbing the tree of nutrients, they have deep roots and you can always feed them if they are fruit trees. I have a Meyer lemon that I have been growing under for years. The thing provides lemons for three families and I rarely feed it. Nastutiums are wonderful, they are pretty and you can eat them too! I agree it's a nice combination.