So, the horse symbolizes Tevye at his most beaten down. At home, though, he's king of his castle (at least in theory). Tevye's house symbolizes his accomplishments, his place in the world, and his family.
Of course, Tevye being Tevye—and Sholem Aleichem being Sholem Aleichem—you know that it's not going to be good for long. The end of the stories brings wave after wave of threats to this house.
First, Tevye sells off his dairy stuff (and with it, memories of the dead Golde) to go on his trip to Israel.
Second, the village mayor and a mob of Gentiles destroy it for the sake of looking good in front of government officials.
Finally, Tevye has to short-sell it to the mayor before he's evicted, along with the rest of the Jews. He comes back to find Tzeitl packing up and the house in a sad state, and this is when Sholem Aleichem really lays out the symbolism:
I came home to find, not a house, but a wreck, the poor walls bare, as if they were shedding tears for all that was happening to them! On the floor were piles, bundles everywhere! On the hearth the cat perched, sorrowful as a poor orphan. […] Here was where I had grown up, here I had struggled all my life, and suddenly—Lech l'cho—get thee gone! Say what you will, it's a terrible loss! (9.54)
Here, the house is personified as "shedding tears"—the tears that Tevye thinks he's too manly to shed, maybe. Not only that, but it's pretty much lost its identity as a home—the floor is no longer a walking surface; the hearth (a.k.a. the fireplace), which would usually be where everyone would gather to get warm and eat food, is now occupied by an "orphan."
As the house dissolves, so does Tevye's identity. He's literally homeless, because his house has been packed up and sold. Know who else is homeless? (At least until 1948.)