Usually, on land that is intended to be preserved, they don't want random people hacking away at vegetation, so they will have rules about it. If a park ranger or someone like that sees you cutting down trees or whatever, you are probably going to get yelled at or fined or something.

It's also highly dependent on species and location. Some invasive species will basically multiply if you try to tear them out, either resprouting vegetatively, or through seed spread. Species like tree-of-heaven or paulonia also become huge trees, so they probably don't want you cutting those down.

Some places like Oahu are basically 95% invasive species, so if you remove that, you have nothing left. Oahu is basically all guava and mesquite trees, and without that, the soil washes away, and there's no hope of recovery, so invasive management needs to be done in consultation with experts.