How do you deal with irrational people in your life?

The most important consideration when dealing with other people (rational or irrational) is to judge where they fit in your personal value hierarchy. Based upon this, you can decide what interactions you're willing to have with that person, and which you aren't. It doesn't make much difference whether your grocer is scrupulously rational so long as the food is good, and reasonably priced. On the other hand, you'll probably require a great deal more for a person to be a close friend. When choosing a spouse, it's important to find someone who is not only rational, but who is also a perfect reflection of your values. In other words, you decide to what extent you're inclined to associate with a person based upon their merit and value to you.

When dealing with people where it's not an easy choice to simply not associate with the person, you still have some choices. If the person is actively harmful to your values, it may be necessary to break off the relationship and bear the consequences. For work, this may mean quitting a good job to avoid an extremely dysfunctional relationship with a boss. For family, this may mean simply cutting off ties with the offending family member: even if that's very painful to do.

However, these are all unusually extreme cases. More often, the person has a number of good points mixed with some bad points. In these cases, you can try to limit your relationship with that person to isolate just the good parts and avoid the others. For example, if your parents have differing political views from yours (mine do), you can simply agree not to discuss politics (we have). If a co-worker just rubs you the wrong way, you can either attempt to isolate your contact with that person to what is necessary, or attempt to find a way to mitigate the behaviors which bother you. For example, I had a coworker a while ago who I felt was consistently attempting to denigrate my work by various verbal and non-verbal communication. I was able to mitigate the situation by getting to know him better outside of work, which in turn, resulted in a more harmonious working relationship.

Naturally, your specific situation will vary, and therefore the best course of action will vary. The basic principle is to identify where that person fits in your value hierarchy, and then restrict the relationship with that person to an appropriate level.