What's your interpretation?
After thinking about it for a while, I got an overall idea but it's full of problems and therefore would benefit from further views. In particular, I find some of Tarr's claims difficult to reconcile with the film itself (such as the insistence on its being a simple tale and on the neighbour's monologue being nothing but drunken nonsense). Conveniently, I am however accepting some of his claims to further my take on it. Anyway, here's my interpretation (I'll put my problems in brackets):
A central theme of the film is "giving up" in the face of special/ unexpected circumstances. At the very beginning, the anecdote about Nietzsche, as Tarr himself has stated, shows the philosopher feeling compassion for the horse, "the weak", thus contradicting and giving up a central tenet of his own philosophy. The theme recurs throughout, ranging from the horse's refusal to move or eat to the final actions of his owners, and is perhaps mirrored, as many have pointed out, by Tarr's decision to stop directing films after completing The Turin Horse.
The increasingly apocalyptic weather conditions are faced by father/ daughter with stubborn adherence to daily routine, and, for a while, no deeper meaning seems to be attached to them. This changes with the arrival of the neighbour, who asserts that the nearby city has, literally it seems, been wiped out, and offers a philosophical interpretation (this is the most important and most difficult part). His words are Nietzschean, and Tarr himself has called the neighbour "a Nietzschean shadow". In particular, he seems to announce the famous "God is dead": the good and noble have disappeared because they've understood that there's neither God nor gods i.e. there are no moral absolutes or metaphysical entities and the world is in a state of passive nihilism (but Nietzsche only speaks about collapse of values not of actual world destruction, so is Tarr depicting the aftermath of God's death metaphorically?). Forces which seem to represent materialism appear to have prevailed (is there a social/political rather than philosophical sense here, as some have argued? In particular, why is world destruction labelled as "man's judgement on himself"?). At the same time, the neighbour is not, as Tarr notes, a pure Nietzschean figure (why does he state that God is taking part in what is happening when he's just said that there is no God? Perhaps he's still uncomfortable with God's death? Perhaps this is drunken incoherence?). In any case, the neighbour's interpretation is dismissed by the father as nonsense, which recalls the similar reaction to the Madman's assertion that God is dead in Nietzsche.
Even after the neighbour's revelation of the state of passive nihilism, the father/ daughter routine continues unchanged. There seems to he no attempt at achieving Nietzschean active nihilism i.e. embracing the current situation with joy as a step towards the status of superman/ overman. This is in contrast with the gypsies, who, Tarr says, symbolise vitality and freedom. They give the daughter a book which Tarr describes as an anti-Bible, in which it is said that churches are to be closed because men have sinned (closure of churches as a further confirmation that God is dead?).
Father/ daughter ultimately break their routine and leave once the loss of water compromises their survival, only to return shortly afterwards. Their routine now becomes overtly a trap from which they can't escape and which, at the same time, becomes increasingly difficult to maintain as the world keeps un-doing in the reverse Creation over six days. In the final scene, the daughter seems to have given up completey and decided to starve herself to death, just like the horse is doing.
The philosophy motivating the actions of father and daughter is what I find most difficult to pin down. Are they Christians who refuse to understand that God is dead and hold on to hope until almost the end? (but there is no sign of religion, which is puzzling in a 19th-century peasant household). Or perhaps they, like Camus' Sisyphus, accept the absurdity of their repetitive, meaningless existence until this pattern is broken? (but why is the pattern broken and the world ending, just an extreme test?). However, their behaviour seems most close to that advocated by Nietzsche's nemesis, Schopenhauer: in the face of passive nihilism, man should try to destroy the will which guides and torments his existence through asceticism, i.e. chastity, fasting, mortification of the body etc. rather than become a superman/ overman. This seems very similar to the minimal existence which father/ daughter lead in the film, in which they appear to have no desires at all, whether to enjoy their food or to have a conversation. Death by starvation, which the daughter apparently chooses, is also approved by Schopenhauer. Following this line of thought, what the film seems to say is that, in the face of nihilism, man's response can't be active and joyous as Nietzsche would have it but rather directed towards ultimate self-destruction. The negation of Nietzsche's philosophy at the beginning would then be mirrored at the end (if this is the case, why do they leave? And why does the father state at the end that they must eat? Is this to show that, despite their ascetic existence, the will to live is still there until the end? Wouldn't this mean that they follow Schopenhauer out of desperation rather than conviction?).
Sorry for the long post.