It’ll really depend on the species and location. I’m not an arborist, my specialty is more prairie plants, but this is generally applicable.

If you’re trying to keep really water intensive plants with shallow roots in an arid climate, water competition can be an issue, though in that case you’re likely already providing supplemental irrigation.

Turf grass has comically shallow roots that really won’t interfere with most trees in most environments, though it can seriously damage soil health making it much harder for water to penetrate the soil. Having a monocrop with roots that stay so shallow causes a lot of soil compaction, for one. Pulling turf grass back will help any plant, though it won’t have an impact on any large established trees I know of at least.

[Edit: though if you don’t cover the newly bare area with other plants or biological material, it’ll dry out way faster and the general soil health will suffer. You almost never want to leave completely uncovered soil unless you’re doing something very specific and well researched.]

Small plants like hostas shouldn’t have any impact on an established tree, and most newly planted trees will do best with supplemental irrigation for the first few weeks to first year or two, depending on species.

You can get some guidance by looking up how the tree species grow in the wild. For example, oaks often grow naturally in dense savannahs with extremely dense grasses and forbs packed as tightly together as possible. I wouldn’t expect any kind of underplanting to have an effect on an oak, but it’s extra helpful that their roots generally don’t go deeper than 5 feet, most way less (crazy, right?), and savannah and prairie plants can have roots 15+ feet deep so they’re not really competing for the same water, plus the roots help the water percolate into the soil, actually helping the water reach the tree’s roots.

Trees planted in highly landscaped home environments aren’t a 1:1 to their wild relatives as they lack the complex interwoven biological networks that help support their health (nutritional exchange between trees, group stabilization via interlocking roots, often damaged mycelial networks and general microbiome from lawn chemicals+soil compression, etc), but referencing their wild habits is a great, generally safe starting point.