Actually the suggestion that the earth is an infinite plane is much much more complicated than the usual flat earth image (e.g., the UN flag map). If the earth stretches out infinitely then the stars, sun and moon really have no place to set or from which to rise. Somehow, the sun goes down (or appears to go down) in the west - but it really cannot go below the horizon because there is no space since the earth is infinite, so somehow the sun is teleported over to the east on the horizon. Of course if I make a long distance phone call to Japan or Australia, someone will there will tell me that the sun is coming up in their east at the very moment it's going down in my west - which shouldn't happen if the earth is flat nor if it is an infinite plane, either way, especially since, when it goes down in their west it will once again come up in my east.
If the earth is an infinite plane, then the entire sky and all that's is in it is an illusion. On the other hand, with an infinite plane, we should be able to explore for new and unpolluted and unexhausted land and water, and I wonder why the countries of the world haven't made the effort.