On lens without the infinity sign indicator, on AF mode, AF on the furthest object (i.e. something on the horizon), not really at inifinity but it's close enough. Switch to MF and shoot away at infinity focusing.
In optical terms:
light from a point source diverges
if that object is sufficiently far away (virtual infinity), the separate light rays from it appear parallel
for objects closer than infinity, the rays are divergent to some extent
'beyond infinity' implies that light rays from a point source are converging on the lens, which doesn't happen in the real world
I believe most lenses can be adjusted to focus 'beyond infinity' -- they are usually adjusted so that infinity is at the max end stop. There doesn't seem to be a reason to focus 'beyond infinity', not for normal DSLR lenses anyway.
I'm not sure if it's necessary for IR photos (dun think so). Manual lenses used to have a different focus index for IR.
Optically, every lens has exactly one plane of focus, a fixed (finite) distance from the lens. Technically speaking, no lens can focus at infinity. The area in front and behind that focus plane deemed to be "acceptably sharp" (but not in perfect focus) is called the depth of field. The "acceptably sharp" area in front of the focal plane is the front depth of field, the one behind the plane is the rear depth of field. As the focal plane moves away from the lens (ie, lens focuses further away) the rear depth of field increases rapidly. Beyond a certain subject distance, everything from the focal plane "to infinity" is considered acceptably sharp, hence the lens is considered "focused to infinity". That focusing distance is also known as the lens' hyperfocal distance.
"Beyond infinity" is built in to allow the lens to focus "to infinity" in temperature extremes when the lens contracts or expands.