Are you sure? I would think that this is what it would be like (if you could see forever).
Horizon on Sphere:

Horizon on Infinite plane (remember, you can see forever):

Yes, I'm sure. It's called the geometrical horizon. Think about it: you can see on forever, but things just shrink more and more as they go off into the distance becoming
infintesimally small as they approach the horizon. In fact, due to the earth's size, the difference between our true horizon and what the horizon would look like if we lived on an infinite plane is imperceptably small. (i.e., they'd look the exactly same to the naked eye). The difference is how objects would behave upon receding into the horizon.
From wikipedia's horizon article:
In many contexts, in particular perspective drawing, the curvature of the earth is typically disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the viewer increases. Note that, for viewers near the ground, the difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is typically imperceptibly small. That is, if the Earth were truly flat, there would still be a visible horizon line, and, to ground based viewers, its position and appearance would not be significantly different from what we see on our curved Earth.