What does the horse symbolise in Batman v Superman: Dawn of Justice?

The horses possibly represent the Four Horsemen of the Apocalypse.

The overall appearances of the horses on different occasaions/events collectively represent the innocent deaths, the chaos, fall of Batman, the rise of Doomsday, Superman's death etc.

This is explained nicely by the user delarozay on Reddit:

Basically the 4 horseman (or horses) of the Apocalypse symbolize the end of times, so I just interpreted the first 4 appearances of horses as the build up to the end. If the 4th horse was in fact the one that was shown after the bomb went off then it makes sense as that would be when it all comes apart. Innocent people die, Doomsday is created, end of times as prophesied. However there probably is a deeper meaning behind each of the times they're shown.

The first horseman of the Apocalypse is mentioned in Revelation 6:2: “I looked, and there before me was a white horse! Its rider held a bow, and he was given a crown, and he rode out as a conqueror bent on conquest.” - The 1st horse we see crosses Bruce's path so this passage ties well with Batman.

The second horseman of the Apocalypse appears in Revelation 6:4, “Then another horse came out, a fiery red one. Its rider was given power to take peace from the earth and to make men slay each other. To him was given a large sword.” - The 2nd horse is shown in Lex Luthor's home, he's our villain and the passage fits him perfectly.

The third horseman is described in Revelation 6:5-6, “...and there before me was a black horse! Its rider was holding a pair of scales in his hand... - The 3rd horse is shown on the wallpaper while Clark & Bruce are speaking with each other ("pair of scales").

The fourth horseman is mentioned in Revelation 6:8, “I looked, and there before me was a pale horse! Its rider was named Death, and Hades was following close behind him. They were given power over a fourth of the earth to kill by sword, famine and plague, and by the wild beasts of the earth.” - The 4th horse is shown after the explosion in the courthouse where at this point in the film we reach our boiling point.

While the existing answers are certainly interesting and Zack Snyder hasn't held back with Biblical references in the past either, you don't necessarily have to dig as deep as the Four Horsemen of the Apocalypse.

If we look at the instances of where we prominently see horses, as denoted in your question, we have the aftermath of the fight with Zod in Metropolis as well as the explosion of the Capitol. The primary aspect that is important when seeing those horses there is, that horses react rather instinctively and also usually have a rider.

A rider-less horse strolling around through the rubble primarily evokes an image of chaos and lost control, a world that is out of order (especially when we consider that those are actually police horses). It is similar to a parent-less child wandering around, there's something missing and we immediately know things are out of control (or how often do you encounter wild horses in your everyday life?). The same goes for the horse that is revolting during the Capitol explosion, it's the loss of control again that is important here and how the horse immediately reacts to show us that law and order are powerless (especially together with the in itself symbolic destruction of the US Capitol).

In fact this motif is even evoked later by noone else than Johnathan Kent (or Clark's projection of him) when he refers to the destruction he inadvertently caused on another farm as a child (a dialogue that is essential for Clark's sacrifice at the end):

We blocked the water, alright, we sent it upstream. All the Lang farm washed away. While eating my hero cake, their horses were drowning. I used to hear them wailing in my sleep.