Tiramōrehu was born at Kaiapoi pā, probably early in the nineteenth century, into a high-ranking family of the prominent hapū Ngāi Tūāhuriri of Ngāi Tahu. His father was Kāraki. Tiramōrehu was a descendant of Tūāhuriri through Tūrākautahi, the founder of Kaiapoi pā. Tiramōrehu's mother was Hinerukutai, of Ngāti Mamoe.
After Kaiapoi was sacked by Ngāti Toa under Te Rauparaha in 1831, Kāraki took a leading part in the retaliatory Ngāi Tahu raids. Tiramōrehu was said to have fought alongside his father and to have been wounded. In 1837, while the feud was still unresolved, Tiramōrehu led a migration of Kaiapoi people south in a flotilla of canoes, to settle near John Hughes's whaling station at Moeraki. His relations with the Europeans at Moeraki were friendly but formal, and his community avoided the dissolute way of life said to have been common among whalers.
On 16 May 1840 the first Christian mission among Ngāi Tahu was established by the Wesleyan missionary James Watkin at Waikouaiti, south of Moeraki. In December 1840 the Roman Catholic bishop, J. B. F. Pompallier, preached at Moeraki, and in 1843 Te Rauparaha's son Tamihana preached there on behalf of the Anglican mission. Both won converts, but on 30 July 1843 Tiramōrehu led his Moeraki followers into the Wesleyan faith, and was baptised at Waikouaiti by Watkin. He took the name Matiaha (Matthias), and became a fervent supporter and lay teacher of the Waikouaiti Wesleyan mission, in company with the chiefs Horomona Pōhio, Natanahira Waruwarutū, Rāwiri Te Maire and Merekihereka Hape.
Tiramōrehu had also been trained in the traditional learning of his tribe. When Edward Shortland visited Moeraki in 1844, Tiramōrehu impressed him with his knowledge of Māori traditions, but declined to discuss them on a Sunday. Watkin noted that Tiramōrehu was 'perhaps better acquainted with genealogical antiquities than any other person', and the missionary J. W. Stack considered him 'the best authority on Māori traditions in the South Island.' Until 1868 Tiramōrehu conducted a wharekura (school) at Moeraki, where instruction in traditional learning was given to young Māori.
Tiramōrehu encouraged his people to adopt European agriculture, and looked forward to their becoming prosperous farmers and pastoralists. They retained customary rights in the Kaiapoi district, which was included by Governor George Grey in the Wairau block purchased from Ngāti Toa in March 1847. Tiramōrehu was present when Ngāi Tahu objections were raised at Akaroa in February 1848, during a meeting with Grey to discuss the purchase of a site for the Canterbury settlement. He later claimed that Grey had misled the chiefs into thinking that their objections to Ngāti Toa claims to Kaiapoi would be heeded by the government. When Henry Tacy Kemp negotiated the Canterbury purchase at Akaroa in June 1848, Tiramōrehu signed the deed with his name Matiaha. It was the only tribal land sale deed he signed.
Later in 1848 Walter Mantell was sent to allocate Māori reserves under Kemp's deed and was hospitably received at Moeraki by Tiramōrehu, who provided him with support and assistance, while remonstrating about the small reserves which Mantell was providing. He also complained that Kemp's payment of half the purchase price to two individuals for distribution to the rest had resulted in the money 'taking wings'. Mantell adopted Tiramōrehu's advice about which chiefs were entitled to receive the remaining instalments for distribution.
Tiramōrehu took a leading part in protests against alleged breaches of Kemp's agreement. In August 1848 he had taken part in a deputation of Ngāi Tahu to E. J. Eyre, lieutenant governor of New Munster, in Akaroa, to request a considerable reserve of lands for Ngāi Tahu. His knowledge of English enabled him to understand fully the purport of Eyre's evasive response, and possibly some unguarded comments between Eyre and Mantell. His letter of 22 October 1849 to Eyre was the first formal statement of Ngāi Tahu grievances about South Island land purchases. Mantell had provided Ngāi Tahu with an average of only 10 acres per head in reserves, although Kemp had promised that reserves would be generous. At Waikouaiti Mantell had provided one European, Johnny Jones, with 2,560 acres, but at nearby Moeraki he had allowed Māori only 500 acres for 87 people, one of the lowest proportions of land to population among Ngāi Tahu settlements. Tiramōrehu's letter is an eloquent and impassioned protest. 'This is but the start of our complaining to you', he wrote. 'We shall never cease complaining to the white people who may hereafter come here.' Tiramōrehu also pursued the promises of schools and hospitals, which had been a condition of Kemp's purchase. He opposed the payment of school fees by Moeraki Māori, on the grounds that they were entitled to free schools.
After his parsimonious allocation of South Island reserves, Mantell had a change of heart, and became eager to assist Ngāi Tahu. On his departure for England in 1855 he sent a consignment of Māori schoolbooks to Tiramōrehu. In January 1856 Tiramōrehu and other chiefs proposed to Governor Thomas Gore Browne at Otago that Mantell should be made governor of the South Island, and in September 1857 they addressed a similar request to the Queen.
In February 1857, at Kaiapoi, W. J. W. Hamilton purchased the North Canterbury block on behalf of the Crown. Tiramōrehu was absent through illness and later complained that the rights of his Moeraki community had been overlooked. Ten years later the government paid him £200 in compensation for this oversight. On 7 September 1860 Tiramōrehu was appointed one of two native assessors for Otago. The position carried the duties of a local magistrate, at a salary of £10 per annum. By 1866 he had relinquished the post, but had resumed it by 1869.
In his old age Tiramōrehu renewed his efforts to obtain more land for Ngāi Tahu, taking a leading part in organising meetings at Otago and Kaiapoi in 1874, and at Arowhenua in 1875, to further the claim. He continued his 30 year correspondence with Mantell, since 1866 a member of the Legislative Council. The culmination of these efforts was the setting up in February 1879 of the Commission on Middle Island Native Land Purchases (the Smith–Nairn commission) to investigate Ngāi Tahu grievances. Tiramōrehu gave evidence to the commission in 1879 and 1880. His testimony is an important source of information on Ngāi Tahu occupancy of the South Island before the arrival of Pākehā settlers.
On 6 October 1849, at Moeraki, Charles Creed had married Tiramōrehu and Pirihira (Priscilla) Pī of Ngāi Tūrākautahi. Matiaha Tiramōrehu wrote his own name in the register in a clear hand. This was his only registered marriage although he had two other wives, whose names are not recorded. Tiramōrehu and Pirihira had two daughters, who died in infancy.
Tiramōrehu died at Moeraki on 7 April 1881. He was said to be over 80 years old. Pirihira had predeceased him by some years, and he left no direct descendants. His tangi occupied a week and drew 500 mourners. He is commemorated in stained glass windows in the Māori church at Moeraki.