I think the better question would be, what are the reasons for accepting the filioque? For a more thorough refutation of the filioque doctrine, I strongly recommend reading Mystagogy of the Holy Spirit by St. Photios, my patron saint and one of the greatest theologians of the 9th century. This is a very complex issues, but I will summarize the main problems as quickly as I can.
The Fathers have all taught that God is one in essence and three in persons. But what is the difference between the essence and persons of God? The answer to this has been unwavering: In the Triune God, the essence is the singularity of God while the persons are the distinctions. To be clear, what the Father, Son, and Holy Spirit share is the essence of God, and is shared equally by all persons. Where the essence ends and the person begins is the unique characteristics of each person. The Son is generated, and the quality of generation is unique of the Son alone. What the Son has in common with the Father must also be common to all three, and thereby of the essence. For if there is a quality shared by two persons with exclusion of the third, then you have the qualifying factor comprising a person. So if the Son and the Father share a quality distinct of the Holy Spirit, then a "person" is manifested. However, there are only three persons, so it remains that the only acceptable Trinitarian method is that of the following: what is common is the essence, and what is distinct are the persons.
So the filioque suggests that the Spirit proceeds from the Son as well as the Father. First of all, there is the obvious problem of persons. If the quality of filiation is shared by both the Father and the Son, then it cannot be a personal quality, but of the essence. But if it is of the essence, then it must also be shared by the Holy Spirit. But this would result in the absurd concept of the Holy Spirit proceeding from Himself. So either the filioque denies the essence/person distinction, confusing the Trinity, or it requires that the Holy Spirit proceeds from Himself, and absurd concept which removes the Father as the "fount of Divinity". Both of which are heretical.
This doctrine also causes problem because while it ignores the essence/personal distinction in suggesting that the Father and Son alone possess the quality of filiation without the Spirit (thereby compromising a Person that the filioquists reject despite the logical requirement to do so), it causes the Holy Spirit to be less in Divinity than the Father and the Son. And I think this point is the least complex and easiest to understand for the filioquist. You have the essence, which all three Persons possess. And then you have the distinctions between the Three that is peculiar to each Person. But the filioque tags on an additional relationship between the Father and the Son that the Holy Spirit is not involved with. So according to the filioque, the Father and the Son have a sub-essence/extra-personal bond that places them closer together in their relationship and applies greater Divinity to them than is given to the Holy Spirit. The Holy Spirit owes its procession to both the Father and the Son, and there is no longer equality of persons in the Trinity.
There are many more problems with the filioque that are every bit as problematic as what I've just shown. Like I said, I suggest reading St. Photios' Mystagogy which treats this issue in great detail. It should be noted that the Fathers have always agreed on the economic procession of the Holy Spirit from the Son. And that is represented by saying that the Spirit proceeds from the Father through the Son. But the West has taken this even further and suggested not only is the procession economical, but eternal -- that is, the Spirit proceeds eternally from the Son, which is a grave heresy. And it is utterly opposed to the monarchia of the Father which has always been the very foundation of Trinitarian theology. Not to mention, the filioque was only introduced into the Creed in the West primarily in Spain to battle lingering elements of Arianism. We are bound to accept the unanimous teaching of the Fathers. The overwhelming popular belief of the Fathers is that the Spirit eternally proceeds from the Father alone.
(note: For those unaware, Athanasius never taught the filioque. Roman Catholics like to cite the Athanasian Creed as evidence that he did. But it has been conclusively proven and accepted by both sides that this was a forged document originating in Gaul over 100 years after Athanasius' death -- written in Latin no less)