I think he liked Tyrion but I in no way think that his main motive for fighting was his fondness for Tyrion.
The whole reason he came to King's Landing was to find out what happened to his sister, who gave the order and if the Mountain did what Oberyn thought he did.
Tyrion Lannister: Why did you come to King's Landing, Prince Oberyn?
Oberyn Martell: I was invited to the royal wedding.
Tyrion Lannister: I thought we were speaking truth.
Oberyn Martell: [pause] The last time I was in the capital was many years ago. Another wedding. My sister Elia and Rhaegar Targaryen, the Last Dragon. My sister loved him. She bore his children. Swaddled them, rocked them, fed them at her own breast. Elia wouldn't let the wet nurse touch them. And beautiful, noble Rhaegar Targaryen... left her for another woman. That started a war, and the war ended right here, when your father's army took the city.
Tyrion Lannister: It wasn't actually...
Oberyn Martell: They butchered those children. My nephew and niece. Carved them up and wrapped them in Lannister cloaks. And my sister. You know what they did to her?
[Tyrion has been staring at the ground uncomfortably for some time; Oberyn lifts his chin with a finger]
Oberyn Martell: I'm asking you a question.
Tyrion Lannister: I've heard rumors.
Oberyn Martell: [chuckles] So have I. The one I keep hearing is that Gregor Clegane, the Mountain, raped Elia and split her in half with his greatsword.
Tyrion Lannister: I wasn't there. I don't know what...
Oberyn Martell: If the Mountain killed my sister, your father gave the order.
Oberyn Martell: Tell your father I'm here. And tell him the Lannisters aren't the only ones who pay their debts.
[Oberyn walks away]
Spoilers about the fight:
Throughout the fight scene he is constantly shouting "Elia Martell of Dorne, you raped her, you murdered her. You killed her children." This is aimed at the Mountain and is also for the crowd. He also looks to Tywin when he says "Who gave you the orders?"
The fight results are featured in this spoiler:
He is so obsessed with getting the Mountain to admit to what he has done that he eventually loses the fight because of this.
In conclusion Oberyn's reasons for fighting were mainly his own for his pride and for his sister and for the house of Martell, however he did also like Tyrion. The opportunity to fight the Mountain arose and clearly he seized it.