I don't think I could improve upon the summary at the Orthodox Wiki:
Objections on doctrinal grounds
- It is contrary to Scripture, particularly in John 15:26: "But when the Helper comes, whom I shall send to you from the Father, the Spirit of truth who proceeds from the Father, He will testify of Me." Thus, Christ never describes the Holy Spirit as proceeding from himself, but only mentions the Spirit's procession in terms of the Father.
- The justifications for including the filioque in the Creed—bolstering the divinity of the Son and emphasizing the unity of the Trinity—are redundant, given the original wording of the Creed. That is, the Son already is described as "light of light, very God of very God," and so forth. The Spirit also "with the Father and Son together is worshiped and glorified." Additionally, the Creed itself begins with a statement of belief in "one God."
- The filioque distorts Orthodox Triadology by making the Spirit a subordinate member of the Trinity. Traditional Triadology consists in the notion that for any given trait, it must be either common to all Persons of the Trinity or unique to one of them. Thus, Fatherhood is unique to the Father, while begottenness is unique to the Son, and procession unique to the Spirit. Godhood, however, is common to all, as is eternality, uncreatedness, and so forth. Positing that something can be shared by two Persons (i.e., being the source of the Spirit's procession) but not the other is to elevate those two Persons at the expense of the other. Thus, the balance of unity and diversity is destroyed.
- Given the previous objection, the repercussions to the acceptance of the filioque into church life are potentially massive. Because how we relate to God is significantly affected by what we believe about him, false beliefs lead to damaging spirituality. One objection often raised about Filioquist theology is that it undermines the role of the Holy Spirit in the Church. Thus, with his role being denigrated, his traditional ministries are effaced or replaced. The Church's unity becomes dependent on an office, spirituality becomes adherence to the letter of the law rather than its spirit, sacraments come to be understood in terms of validity, and a spirit of legalism prevails.
Objections on canonical and historical grounds
- Though not really a question of heresy, a common objection is to the means of inserting the filioque into the Creed. That is, unlike the original adoption of the Creed at Nicea and its subsequent revision at Constantinople, the decision to include the filioque in the Creed was not done by an Ecumenical Council. Rather, it was initially inserted by the Third Synod of Toledo, Spain (589).
- Rome resisted the inclusion of the filioque for centuries. Leo III, the Pope of Rome at the time the filioque began its history in Western theology, strongly advised against its inclusion, even though he agreed with the soundness and validity of the doctrine contained in filioque. Later, however, Rome contradicted its previous more Orthodox stance by the promulgation of the filioque, thus anathematizing its own spiritual forebears.
Here's more on the ecclesiastical issue:
For Roman Catholics the Nicene Creed is under the Pope, not over the Pope. When the Pope inserted the Filioque into the Nicene Creed a major realignment of ecclesial authority took place. The Pope without the assent of the other historic patriarchates: Constantinople, Alexandria, Antioch, and Jerusalem, and without convening an ecumenical council of bishops, unilaterally altered the Nicene Creed. This was done even though the Third Ecumenical Council (Ephesus 431, Canon VII) forbade the creation of a new creed. In essence, the Bishop of Rome was claiming a magisterium (teaching authority) equal to or superior to the Ecumenical Councils. In exerting authority over the first three Ecumenical Councils the Pope was claiming authority over all Seven Ecumenical Councils. Simply put, the Bishop of Rome, once first among equals, now claimed supremacy over all Christians, a startling departure from Tradition. The emergence of a papal model of authority would in time clash with Orthodoxy’s conciliar model of authority. Here we see how the Filioque lies at the root of the West-East Schism.
The Eastern Orthodox view, in contrast to the filioque, is that of the "monarchy of the Father." Here it is described by Fr. John Behr:
According to the Nicene creed, the Son is “consubstantial with the Father.” St Athanasius, the Father who did more than anyone else to forge Nicene orthodoxy, indicated that “what is said of the Father is said in Scripture of the Son also, all but His being called Father” (On the Synods, 49). It is important to note how respectful such theology is of the total otherness of God in comparison with creation: such doctrines are regulative of our theological language, not a reduction of God to a being alongside other beings. It is also important to note the essential asymmetry of the relation between the Father and the Son: the Son derives from the Father; He is, as the Nicene creed put it, “of the essence of the Father” – they do not both derive from one common source. This is what is usually referred to as the Monarchy of the Father.
St Athanasius also began to apply the same argument used for defending the divinity of the Son, to a defense of the divinity of the Holy Spirit: just as the Son Himself must be fully divine if He is to save us, for only God can save, so also must Holy Spirit be divine if He is to give life to those who lie in death. Again there is an asymmetry, one which also goes back to Scripture: we receive the Spirit of Him who raised Jesus from the dead as the Spirit of Christ, one which enables us to call on God as “Abba.” Though we receive the Spirit through Christ, the Spirit proceeds only from the Father, yet this already implies the existence of the Son, and therefore that the Spirit proceeds from the Father already in relation to the Son (see especially St Gregory of Nyssa, To Ablabius: That there are not Three Gods).
So there is one God and Father, one Lord Jesus Christ, and one Holy Spirit, three “persons” (hypostases) who are the same or one in essence (ousia); three persons equally God, possessing the same natural properties, yet really God, possessing the same natural properties, yet really distinct, known by their personal characteristics. Besides being one in essence, these three persons also exist in total one-ness or unity.