It's kind of hard to answer this definitively, studies of Manichaeism are still somewhat limited and our sources are few. What we do know about it often comes from its opponents, especially the former Manichaean Augustine of Hippo, who would convert to Christianity. Recently, we have found a number of Manichaean texts in a variety of languages, commonly in Syriac, but sometimes in Chinese, Uyghur, Sogdian, Greek, Latin, Parthian, or various Persian Dialects. These have been remarkably useful in deciphering Manichaean Cosmology and Philosophy, but our picture is still very much incomplete. The exact number of intact sources escape me, but I believe it to be in the mid-hundreds, surely a fraction of the volume of work produced which must have been in the thousands, at the very least. We also have visual sources in iconography and paintings that survive, though they are also curiously small. Suffice to say that the purging of Manichaean sources by persecution and the ravages of time have left us with less ability to judge it's influence than other religions.
Compounding this is that Manichaeism, like many other religions that spread over a long geographical area in that era, was remarkably syncretic. While Manichaeism itself was primarily rooted in Judaeo-Christian Gnosticism, it absorbed aspects of Zoroastrianism and Buddhism into it during its formative years. Manī, the founder of Manichaeism lived in the Sassanian Empire and traveled through it from modern day Iraq to modern day Pakistan. In his travels, he almost certainly encountered Buddhism as he made Buddha prophet in his doctrine and took the monastic element from Buddhism. After Manī's death, Manichaeism spread into Rome where it became one of Christianity's main rivals, here, it remained rooted firmly in Judaeo-Christian Gnosticism, but it also traveled into Central Asia and China. In Central Asia and China, it absorbed the local gods and customs and took on more and more features of Buddhism. In China, Manī was known as the "Buddha of Light" and throughout Sogdia, Parthia, and the Uyghur Khaganate it existed in competition with Buddhism and was almost indistinguishable. Case and point, after the Tang persecutions of Manichaeism, the Manichaeans were more or less absorbed into Buddhism.
And there's the most interesting bit of all of this. Manichaeism is often thought to be extinct, but it really isn't. Manichaeans still exist, but as a quasi-Buddhist sect. There are a few shrines still active in China, all of which double as Buddhist ones. However, there is are active cults still dedicated to Manī, though their beliefs have become closer to Chinese folk religion.
With all of the above, it can be difficult to gauge how exactly Manichaeism has influenced other religions, but we can certainly try.
Perhaps the most obvious influence is on Chinese Salvationist religions. The first one that most people know of is the White Lotus. The White Lotus first came to be known during the Song Period and were a distinctly Milleniarian sect who worshiped a mother goddess and awaited the coming of Maitreya, the future Buddha. They took certain doctrines and practices from the Manichaeans, who in China had become known as the "religion of light." All members of the White Lotus were vegetarians and practiced Pure Land chanting and lotus burning. The strict vegetarianism and wearing of white robes came from Manichaeism. They were instrumental in Red Turban Rebellion and driving the Mongols from China. Strangely, they were persecuted under the Ming and Qing, leading intermittent rebellions and absorbing more Manichaean doctrines such as the importance of "light" as a concept.
Other Maitreyan salvationist doctrines may also have absorbed Manichaean doctrine, but I cannot comment further as studies of Chinese Salvationist and Maitreyan beliefs are still in their infancy and due to the great variety of beliefs and practices present in China, it is unlikely there will ever be a complete categorization of all of them. That said, it is highly likely that Chinese Folk Buddhism and Folk religion have absorbed much of Manichaeism into itself, as they have absorbed Daoism and every other religion.
There are also less obvious influences on Buddhism. The Buddhists in China were notorious plagiarists (as was every other religious group in China) and they adopted a few Manichaean Sutras and themes into their Sutra literature. Much of this can still be found in the sprawling Taisho Tripiṭaka. Some of these are attacks or commentaries on other religions, but some, such as the Uyghur Sākiz Yūkmāk take distinct themes from Manichaean cosmology. However, the most prominent institutional influence was a reassertion of the non-dual nature of Buddhism in contrast to the dualist nature of Manichaeism.
For the most surprising manifestation of Manichaeism, we look to Modern Christianity. Modern Christian Philosophy has had few more important influences than Augustine of Hippo. His Confessions and City of God adapted Neo-Platonic principals and fused them with Christianity as well as addressing the nature of the soul, ethics, and the problem of evil. Few would deny the Platonic and Neo-Platonic influence on Augustine, but less ackgnowledged is the Manichaean influence. In his youth, Augustine was drawn to Manichaeism for it's philosophical value and its clever answers to the problems of "sin" and "evil" through its dual nature. Augustine turned away from his mother's religion of Christianity and followed Manichaeism having found the answers he wanted in it. In the middle of his life, he converted to Nicene Christianity and became its most prominent exponent. However, Manichaean pessimism remained in his works. Augustine asserts that evil is passed through the line of Adam, that original sin has created wickedness present in all. In his Epistle to Paulinus, he asserts that mankind as a whole, is wicked, which he expands on in the Opus Imperfectum, stating that only through the grace of God are the wicked allowed to be good. Furthermore, his interpretation of human history in The City of God reveals a dualist view of human history in the struggle of man to be light and good and free themselves from corruption and misery.
Sources
Van Oort, Johannes. "Augustine's Manichaean Dilemma in Context." Vigiliae Christianae 65, no. 5 (2011): 543-67. http://www.jstor.org/stable/41480508.
Scott, David. "Buddhist Responses to Manichaeism: Mahāyāna Reaffirmation of the "Middle Path"?" History of Religions 35, no. 2 (1995): 148-62. http://www.jstor.org/stable/1062694.
Buonaiuti, E. "Manichaeism and Augustine's Idea of 'Massa Perditionis'." The Harvard Theological Review 20, no. 2 (1927): 117-27. http://www.jstor.org/stable/1507588.
Miereki, P., & Bedhun, J. (Eds.). Emerging From the Darkness: Studies in the Recovery of Manichaean Sources (Vol. 43, Philosophi Antiqua). Liden, Netherlands: Brill. (1997).
Lieu, Sameul N. Manichaeism in Central Asia and China. Vol. 45. Nag Hammadi and Manichaean Studies. Leiden, Netherlands: Brill, 1998.
Dan, Jennefer Marie. Manichaeism and Its Spread into China. Master's thesis, University of Tennessee-Knocksville, 2002. Accessed May 11, 2018. University of Tennessee Creative Exchange.
Tardieu, Michel. Manichaeism. Translated by M. B. DeBevoise. Urbana, Ill: University of Illinois Press, 2008.