Light that travels eternally because of inflation? What happens with electromagentic waves that are emitted into 
intergalactic space, but reach not any object because of inflation? Lets assume that photon A and photon B are emitted. Photon B hits some object, photon A enters a gigantic void or simply has nothing in its path, and heads for the horizon, so to say.

Relativity tells that A and B do not "feel" time passing. From our point of view B travels a finite time, while in the frame of B the object is reached instantly. However, since A will be eternally underway, what is going to happen? We have kind of a zero*infinity travelling time in the frame of A.

Do our models make statements about this? Or is there some explanation
Infinities are a type of singularity. As soon as you introduce an actual infinity into any kind of mathematical model you can't expect it to make useful predictions. Mathematical models can't deal with infinite time. Similarly, using a frame that moves at the speed of light, with infinite rapidity, is singular.  

In the particular case you describe, in a regular frame, the photon becomes progressively redshifted by the expansion of spacetime, its energy (to an observer in a inertial frame) becomes lower and lower.