I'm not clicking that link for a lot of reasons; if that invalidates my post, scoll along, lil' doggies.
There's a couple problems with incest. The obvious one is inbreeding, but that obviously only applies when offspring are produced.
The other one, in the shortest terms, is that love, sex, and bonds of trust are insanely complicated things. I can't explain every facet of this but I'll go into a couple.
Something that may seem obvious, but that people don't seem to get, is that once you have sex with someone, you can't undo that. Your relationship goes from one that did not involve sex to one that does. Sex does major big things to your brain, carves connections and draws associations that--for the most part--you just have to live with for the rest of your life.
This is one of the reasons why rape and molestation are so awful, is because they associate sex with things like violence and pain and feelings of helplessness and degradation.
So when you have the sort of intimate and unconditional lifelong relationship that family ideally has, a relationship that is typically characterized by trust in a lack of sexual/romantic intent, it introduces a number of complications. This person is going to continue being your family. They share bonds with all of your other family members in this way, and such a relationship will alter both of your relationships with every other family member. What's more, unless this ends up being a lifelong partnership (which has all the other problems multiplied) it's going to severely impede your ability to have relationships with others, as your familial relationship doesn't disappear, and--as said earlier--you can't undo having sex with them.
A lot of the problems that arise from this are problems that arise from having sex with multiple people anyway, but the problems of trust and broader implications for someone's relationships with others are huge, and undoubtedly bigger than I am even aware of or can comment upon from an outside perspective.