Is it permissible for a person following the Hanafi school of teachings to perform his/her Salaah at the earliest possible time (like the shafi's)?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

A Muslim is not obliged to abide by a given school of jurisprudence to the point that he never gives it up; rather, his reference should be the Book of Allaah and the Sunnah of the Prophet in his acts of worship and matters of life.

Imam Ash-Shaafi'i said: "All the Muslims have agreed that if a Sunnah of the Prophet has been confirmed to anyone, it is not permissible for him to leave it and act according to a statement of any person [no matter who he might be]."

Besides, Imaam Abu Haneefah said: "If a Hadeeth is reported from the Prophet we accept it with pleasure and without any objection; and if a statement is reported from the Companions then we also accept it with pleasure and without any objection, but if a statement is reported from the Taabi'is (the generation which followed the Companions) then there is no no difference between us [we are equal] (in this regard)."

Moreover, it is reported in Rawdat Al-Taalibeen that Abu Haneefah was asked: "What about if you say something that contradicts the Quran? He said: "Leave my saying and adhere to what the Quran says." He was asked again: "What about if your saying is against the saying of the Prophet ?  He replied: "Leave my saying for the saying of the Prophet ." Then he was asked "What about if your saying contradicts the sayings of the companions? He said: "Give up my saying for their saying."

There is much evidence in the Quran and Hadeeth that proves that performing the prayer at its early time is preferred except 'Isha and Thuhr prayers when there is  extreme heat.

Allaah says (what means): {And march forth in the way (which leads to) forgiveness from your Lord.} [Quran 3:133]"

Allaah also says (what means): {It is these who race for the good deeds} [Quran 23:61]"

Allaah further says on the tongue of Prophet Moosa (what means): {...and I hastened to You, O my Lord, that You might be pleased.} [Quran 20:84]

No doubt that performing the prayer at its early time is hastening to get Allah's pleasure and towards good.

Besides, in an authentic Hadeeth, the Prophet was asked about the best deeds, and he mentioned amongst them the prayer at its (early) time.

Moreover, 'Aa'ishah, may Allah be pleased with her, said: "The Prophet never prayed an obligatory prayer twice". [At-Tirmithi and Al-Albaani]

As regards the 'Isha prayer, it is better to delay it as long as this does not cause hardship to the praying people. This is from the Hadeeth reported by 'Aa'ishah, may Allah be pleased with her, who said: "One night the Prophet prayed the 'Isha prayer after most of the night had gone and most of the people in the mosque had fallen asleep. Then he came out, prayed and said: "This would be the proper time if this would not cause hardship for  my nation."

As for Thuhr prayer, the Prophet said: "If it is too hot, delay the Thuhr prayer till it gets cooler, for the severity of heat is from the increase in heat of Hellfire." [Al-Bukhari]

In brief, performing the prayer at its early time is desirable whether one follows the Shaafi'i or the Hanafi School of jurisprudence.

Finally, beleiving that it is forbidden for a person who follows the Hanafi School to act according to the Shaafi'i School is a wrong beleif as there is no evidence for it.

Allah Knows best.