MPAs can have effects on the food web. If, for instance, there's a lot of fishing of the top-level predators, they are likely to recover when the fishing pressure is taken away, and then they eat the smaller fish. You might then get a very different food web inside a protected area from the one you see outside. You may get less abundance of the mid-level fishes than you would see outside. That's not necessarily a detrimental impact of an MPA. I would say it just shows us how much fishing has changed the actual structure of our marine ecosystems.
In terms of invasive species, we don't know as much. Generally speaking, ecosystems that are healthier tend to resist invasion more than ecosystems that are more disturbed. It might, then, provide a beneficial impact, but it's not going to prevent invasive species either.