The title really says it all. Would we be able to feel anything if time started running backward?
I don't know what it means for time to be running backward. Time doesn't run and it is going nowhere. The idea of time running backward comes from the scientific representation of time as a numerical variable, and of the passage of time as an increase of the value of this variable. This suggests a scenario where this value would instead decrease. However, science itself seems to take time as a sort of resultant, a function of other things, rather than as a fundamental property of the universe or even of reality.
Essentially, an objective time would be relative to the system considered. According to this, time would have to be construed as a function of the evolution of this system. The consequence is that the time relative to a system would go backward if the sequence of evolution of the system was inverted. Instead of evolving from the past toward the future, the system would be evolving from the future towards the past. The system would go through the same succession of states, except that it would go from future states towards past states. The sequence of states would just be inverted. This is relevant to the physical time of the system.
The question, however, is what it would do to us?
In terms on the physical states of our body, we would go through the same sequence of states, only going from future states to past states: Undying first, being unborn last. But what would that do to our perception of reality?
We seem to have two senses of time. The sense of time relevant to the question seems to be a function of our memories, and our memories seem to be a function of the state of our body. If the physical time of our body was going backward, our body would go through the same physical states, only in an inverted sequence.
Thus, subjectively, we would go through exactly the same sequence of memories. The consequence is that at any one time, we would have exactly the same subjective sense of time as we do in actual fact as time is supposedly going forward. Thus, we would think that time is going forward. Thus, time going backward would make no difference to our subjective sense of time.
We may have a second sense of time, more local, so to speak, whereby our brain perhaps produces our impression of the present, i.e. our sense of "now". If this second sense of time is really a cognitive function of the brain, the effect of physical time going backward would also be as nonexistent as for the first of our senses of time.
However, it is unclear that this is really the case and so, maybe, just maybe, we would notice, through this second sense, the actual direction of physical time.
Yes, we could. In addition to the above, this has a cognitive aspect. The usual illustration of entropy is a" disordering" process that can be visualized relative to what we are used to seeing, a bit like Hume's idea of induction as our habituation to the sun rising daily, that's it.
So reversals of entropy are sometimes illustrated by coffee and cream "unstirring" itself or the dropped egg reassembling, or Christ rising from the tomb. Could happen according to Boltzmann's entropy, but vanishingly unlikely in the world we inhabit under the physical constraints we are "used to."
So if this reversal had a sudden beginning, the sight of a broken egg reassembling would certainly startle us and likely be perceived as time "going backwards," just like it does in reversed movies, which we have no problem comprehending, or Martin Amis's great novel "Time's Arrow."
Now, as others have pointed out, we mightn't notice because our formed memories are being erased. But I believe there is evidence that these events do not happen at the same rate. We could perceive and grasp these startling events, for a "period of time" at least, before memories of many similar events disintegrate.
To be honest, I'm not sure about the logic of this answer. It seems a little too simple, but I don't see problem with comparing the miraculous event seen "as if in a backwards film" with memories of similar events we are used to going in the opposite direction, assuming our brains manage some sort of overlapping layers of duration.
Incidentally, the artist William Kentridge has made movies in which everyone moves backwards, then are run backwards. The effect is weird, you know something is wrong but don't know what. Oh wait! I guess I'm not thinking of the reassembling egg being "unperceived," though not sure what that means.
'Going' backwards in time, would subjectively appear exactly like going forwards in time, because time IS the spreading out of information, so to 'go' backwards would be to withdraw information about you from the world, and information about the world - ie, forgetting. This is premised on the idea such 'going' is even possible, which implies time as a dimension, also called eternalism, or the set of 'B series' theories of time (McTaggart). Also, continuous 'movement'.
In the case of presentism, or (broadly) 'A series' theories of time, there are a set of conservation laws acting as a function. Time going backwards would violate these, so it's like asking what would happen if you reversed gravity - a thought experiment, not really meaningful in itself.
If you could perceive the difference between matter and antimatter directly, maybe. Something about the math makes it so an antimatter universe would be like a matter universe going the opposite direction in time. There's even at least one serious physicist who interprets the Big Bang as akin to a particle collision event, with the imbalance between matter and antimatter in our universe being the result of a temporal symmetry that reflected the "missing" antimatter "backwards in time" around the origin (the Big Bang).
However, antimatter reflects photons in the regular way, more or less, so I don't know that antimatter can be "directly perceived" as such (you can directly see it, in principle, but not its quality as antimatter).