Yes, it is true that Mahabharata starts with a dog. The story is mentioned in Adi Parva, Section 3 (Paushya Parva) as follows:
Sauti said, "Janamejaya, the son of Parikshit, was, with his brothers, attending his long sacrifice on the plains of Kurukshetra. His brothers were three, Srutasena, Ugrasena, and Bhimasena. And as they were sitting at the sacrifice, there arrived at the spot an offspring of Sarama (the celestial bitch). And belaboured by the brothers of Janamejaya, he ran away to his mother, crying in pain. And his mother seeing him crying exceedingly asked him, 'Why criest thou so? Who hath beaten thee? And being thus questioned, he said unto his mother, 'I have been belaboured by the brothers of Janamejaya.' And his mother replied, 'Thou hast committed some fault for which hast thou been beaten!' He answered, 'I have not committed any fault. I have not touched the sacrificial butter with my tongue, nor have I even cast a look upon it.' His mother Sarama hearing this and much distressed at the affliction of her son went to the place where Janamejaya with his brothers was at his long-extending sacrifice. And she addressed Janamejaya in anger, saying, 'This my son hath committed no fault: he hath not looked upon your sacrificial butter, nor hath he touched it with his tongue. Wherefore hath he been beaten?' They said not a word in reply; whereupon she said, 'As ye have beaten my son who hath committed no fault, therefore shall evil come upon ye, when ye least expect it.'
"Janamejaya, thus addressed by the celestial bitch, Sarama, became exceedingly alarmed and dejected. And after the sacrifice was concluded returned to Hastinapura, and began to take great pains in searching for a Purohita who could by procuring absolution for his sin, neutralise the effect of the curse.