How much of the Universe is invisible to us, and how does it affect our theories? I watched an interview with Neil deGrasse Tyson on The Late Show with Stephen Colbert where he briefly mentioned a disturbing thought. It is something that I have never thought of before, and the idea borders on the eerie.

The idea is this (and here comes a mouthful): If all our knowledge of the Universe is based on observations of distant galaxies (and Milkyway stars), and we accept that the Universe is expanding at such an accelerating rate that at some point in the future only our nearest stars will be visible to whoever inhabits Earth, and that then those future inhabitants will need to explain the Universe based only on what they can observe, then it is possible that we are today trying to explain the Universe based on largely incomplete data.

Like, imagine that there is much more stuff out there, stuff that could help us explain some of the scientific "mysteries" such as dark energy/matter, but it is beyond our horizon. We don't know what is missing from our view.

And so I guess the question is this: How does this affect the validity of our current models of the Universe?
It well might be the case that universe beyond our observable universe is filled with pink flying elephants, but this just seems highly unlikely. 

Our current models are usually based on the fact that universe is isotropic, ie there’s nothing special about our observable universe vs everything else there is. Obviously there are smaller and larger deviations, but  on larger scales these are thought to cancel out. This means the simplest position here is to assume all of universe looks more or less the same.

While we can’t currently / no longer observe what is going on on the other side of horizon, there is some evidence supporting above view. Firstly, cosmic microwave background measurements seems to support isotropic universe. Secondly, inflation models seem to provide explanation for isotropic
universe via early causal connection beyond our current observable universe, and as our early universe model is able to explain many other things to great detail, it gives some confidence to this claim as well. 

Since early universe models do support the view universe outside our observable universe is pretty similar to what we can see now, this helps us to rely on observable universe for overall model building. 

Also, there does not seem to be any large mass concentrations outside the observable universe, since these would be seen as bias for movement of galaxies and galaxy groups, which also supports that the rest of universe is at least very close to what we can observe directly. This gives further confidence that our current models are probably not affected much by things beyond our currently observable universe. 

However, since everything outside our observable universe is now causally not connected to us, anything that might be going on over there does and will not affect us - thus it is not necessary nor valid to include any of that in our models either.

To summarize my answer: no, there’s probably no effect on universe beyond the observable horizon that would need to affect our models.