I think Kate's answer covers this nicely, but I just saw an Israeli film called Footnote which, IMHO, does exactly what you're asking about. There are great examples here for your question (and in line with Kate's answer). Spoilers for the film (highly recommended, I'll add) follow.
The film is about a father and son, both of whom are professors in Talmud scholarship (to most people - an obscure academic niche; not even the religious study of the religious texts, but the academic study of them). The father is introverted, meticulous, didactic; he's been snubbed by the academic community because his single, monumental life's work was superseded by somebody else's fluke discovery. The son is outgoing, popular, something of a ham; he's charismatic, but lacking in true gravity. The movie is about the tension between father and son - the father is intensely wounded by the community's treatment of him; he is jealous of his son's success, and sees the community as superficial and worthy of scorn.
So most of the film revolves around the obscure, the procedural, the political. Who got what prize when; who gets to vote on adding new members to certain organizations; all in areas that practically nobody cares about. Here are a few moments the film achieved that had amazing tension:
Reactions to the politics: The film opens with the son being honored in a ceremony, and giving a speech. We don't know any of the background yet. For the entire scene, all while the son is talking, the camera shows us only the father's face. The son is talking about what a brilliant educator his father is, what an inspiration he was; the father's face grows more and more sour. We don't even know what the tension is about yet, but we know this much: the father is furious; and if he's furious when his son is praising him, that praise can't be sincere or authentic - the father is angry; the son is lying. The formal, ceremonial event is very dull, but people's reactions to the event make the scene exciting.
Stakes of the politics: The key conflict in the film comes when the father is erroneously informed he's to be honored with the coveted Israel Prize - actually, it's his son it's meant for. Immediately, his entire life brightens up - he becomes more friendly and more forthcoming; he's proud of himself; we see that the recognition is immensely significant to him. That makes it clear to us what the stakes are - not the dry result of the procedure, but what it means to the characters.
Procedure as metaphor: After some twists, the son bargains with the prize committee to give the father the award, to keep the mistake a secret. One of the caveats is that the son himself needs to write the alleged "judges' comments" recommending the father for the prize. The son sits at his computer, copying the judges' comments recommending him, and trying to fit them to his father. They don't fit - and he keeps changing effusive words of praise to more cautious ones. The formal process demonstrates the distance between the two characters; it's practically a metaphor for how the son wants to appreciate his father, but doesn't, and perhaps can't.
Politics as a tool to be used: Meanwhile, the father is exulting in his newfound glory - obscure though it may be. When a newspaper reporter interviews him, he does not hesitate to castigate the community that has snubbed him for so long - and he pointedly includes his son in his criticism. Here, the formal procedure is a tool for furthering the characters' personal agendas. Procedure is what they're all swimming in, so it can be intentionally and directly used for personal aims.
Hope this is helpful :)