Not necessarily. UAE has around an 88% foreign-born population. Qatar is similar. Both pretty rich countries. (Sure, oil helps.)
But then you've got Luxembourg, Singapore and Switzerland, to name just three. All high levels of foreign-born residents. All in the top economies. Even the US has around 14–15% foreign-born residents, and last I checked it's not one of the world's poorest countries either.
On the other hand, you've got North Korea and Afghanistan. Both extremely low immigration rates. Both pretty low on the GDP rankings.
Of course this is a gross oversimplification and there are multiple other important factors for their economic situations, but just to point out that high immigration doesn't automatically mean weak economies, or vice versa.