My friend Bill recently addressed the question: "What practical difference does a belief or non-belief in the Athanasian doctrine of the Trinity make?"
Bill's answer seems to be: not much, since Christianity is primarily about loving God and neighbor, while belief in the Trinity, especially if overzealous to the point of motivating anathemas, not only does not help but may even inhibit doing both of these things.
There is much that is wrong with Bill's answer, I think. The first problem with what Bill says is that he evaluates things from a highly moralist conception of Christianity that is more at home in 19th century European liberalism than in the historic Christian tradition. Certainly the bishops gathered at Nicaea thought more was at stake in the questions revolving around the Trinity than Bill seems willing to grant; but of these two groups, who is a better judge of what counts as Christian?
The ancient thinking on this matter was something like this. Our union with Christ through baptism and the Eucharist is supposed to divinize us, as is the Holy Spirit. But Christ and the Holy Spirit cannot deify us if they are not divine. Consequently they are divine. But they are named separately and treated as (to some extent) separate entities, for example in the baptismal formula: "In the name of the Father, and of the Son, and of the Holy Spirit." Therefore they are not the same. Of course there is one God but this one God exists in three: the Father, the Son, and the Holy Spirit.
The doctrine of the Trinity is an attempt to be faithful to what God has revealed to us about himself through the scriptures and the apostolic teaching. We affirm one God, but grant a distinction of persons. We don't claim to understand how this can be, but at the same time, there is no reason to assume that we can understand everything that's true anyway. What we cannot do is simply throw out divine revelation because it does not mesh nicely with our modes of understanding.
Of course, much more could be said, but I think at least the following counts as a practical consequence of refusing to believe the doctrine of the Trinity: you are refusing what God has revealed about himself and the authority granted to the church to make authoritative statements of dogma. Both of these refusals are made, furthermore, on the basis of the judgments of one's own reason, which is a fine if you are a Kantian, but Christianity is not Kantianism.
Bill accuses theologians who love their pet explanations of dogma too much of being high-level idolaters. But the charge may be put the other way: the person who refuses to submit himself to the church's teaching and God's self-revelation in scripture because he does not understand it is an idolater, worshiping his own reason and understanding as ultimately authoritative. Bill's treatment of the matter assumes that the doctrine of the Trinity is not revealed, and that the church is not an authoritative body through which God can provide these revelations. This begs a major question, since the vast, vast majority of Christians do not accept Bill's Kantian conception of Christianity.