Relationship friction, moral dilemmas and temptations, power struggles, ego-related conflicts

Having political views that are oppositional to someone else's
Clashing religiously with another person
The character and their spouse having divergent parenting styles
Being teamed with someone whose work ethic is vastly different from the character's
The character's moral code clashing with another person's
Living with someone who has different priorities about work, leisure time, family, etc.
The character's civil liberties conflicting with a governing body's actions
The character's outlook on vaccinations and medical intervention diverging from the culture's
The character trying to work with someone who has a drastically different perspective 
A parent and child having disparate education priorities
Being conflicted over state and moral responsibilities

The character being judged (for how they parent, their politics, not going to church, etc.)
The character being told they're wrong
Being teased
Avoiding topics of conversation around certain people
The character having to tolerate digs about their views to avoid arguments
Other people trying to push their ideas onto the character
The character being pressured to change their beliefs
Needing to lie about a particular view to keep the peace
Having to compromise to avoid strife
Getting into fights or arguments
Inadvertently offending someone 
The character being denied access to opportunities or information because of their beliefs
Not being invited to social gatherings