Several of my travel companions have shown messages of approval/disapproval in response to certain in-game and dialogue actions. What is the point or what are the consequences of their approval/disapproval? Will they do something differently (e.g., leave) if they go to either extreme? Will their feelings affect the storyline or quest availability at all?
Maxing out approval from a companion will grant you a perk, the details of which are listed here. Additionally, several companions (Just about every Human or Synth other than Deacon) can be engaged in a romantic relationship as well, with a few difficult charisma checks as you gain approval. After entering a romantic relationship with a companion, when you sleep in a bed near them for 8 or more hours, the usual Well Rested XP Bonus of 10% is replaced by Lover's Embrace, a 15% XP Bonus.
If you max out your companion's like for you, you'll get a permanent perk. Once maxed out and after you've successfully completed the romance charisma checks, if you sleep while they're your companion you'll get the Lover's Embrace bonus instead of Well Rested.
If you upset your follows enough they'll refuse to travel with you.
Those perks tho!
Maxing out a companion's attitude toward you grants a permanent perk that varies by companion and is retained even if you then dismiss the companion. Do followers in Fallout 4 have any perks?
Companions will refuse to travel with you, per Kalina's answer, but I'm unaware of any situations where they'll refuse you quests or become hostile. I'll edit this answer if I find out I'm wrong on these assumptions.