Short answer: significant, sometimes outright, usually as a marker of difference that was expressed in other ways.

To answer your question, we have to recognize first that 'religion' is not a very stable category, particularly outside western Christianity. As a result of the wars of religion in Europe and historical conflicts over the status of the supranational church, the European Enlightenment produced a conception of the separate spheres of "religion" and "ordinary life" that seem familiar to most people in the west today (and ultimately led to the concept of separation of church and state). Because most of the rest of the world does not have that same particular history (especially since Christian thought of that period embraced a number of concepts that lend themselves to separating a "spiritual self" and "secular self"), trying to separate out "religion" from any number of other cultural practices often is difficult, and it is particularly so in Sri Lanka/Tamil Eelam.

By far, the largest religious group on the island for many centuries has been Buddhists. Indeed, Sri Lankan Buddhism possesses a particular conception of their group's importance through traditions that the Buddha both visited Sri Lanka during his lifetime (scattering Buddha relics throughout the island) and intended Sri Lanka to be a sanctuary for Buddhism should it die out in the mainland. This is recorded in one of the oldest texts in the region, the Mahavamsa, which bills itself as a chronology of the kings of the island, but begins with the death of the Buddha and the sanctification of Sri Lanka. The text charts the rise of several prototypical kings (Asoka in India and later Devanampiyatissa, the first Buddhist king of Anuradhapura) whose stories forward a theory of particularly Buddhist kingship. This is tied to the particular heritage of the island: even as the Dhamma languished in India, Sri Lanka was said to remain a haven for Dhamma. Even the name of the island indicates this heritage: Sri is an honorific that is usually translated into English as "Lord," and lots of important Buddhist entities are given the term. So the very name ascribed the island is one of veneration and sacredness (in the literal sense of "set apart-ness"). How could 'religion' not be a part of the conflict (if one grants that Buddhism is a religion)?

The issue is that Buddhism is expressed as more than meditating for Sri Lankan Buddhists. It entails a whole suite of cultural practices, the most important of which is probably language. Sinhala/sinhalese is the language spoken by the majority Buddhist population and is usually the way in which that group is referred to, rather than Buddhist. In contrast, the second-largest group is Tamil-speaking. Sinhala is a modern language descended from Pali, the Indo-European language in which much of the ancient Buddhist texts were written (rough analog to the West: Latin, with which Pali even shares some interesting structural characteristics), whereas Tamil is a Dravidian language (different grouping of origin languages) spoken mostly by the Indian Tamil ethnic group in far southern India (very near to Sri Lanka). Much of this linguistic grouping is geographical: Tamil speakers mostly live in the north of the Island, while Sinhala speakers live in the central and south of the Island, including the capital Colombo. With such real social segregation, other assorted cultural practices signifying difference inevitably took root and gave both groups a sense of difference.

Despite this ancient heritage, we should be careful to buy into the "ancient hatreds" thesis that motivates a lot of Western understandings of conflicts outside the West (Israel/Palestine, Sunni/Shi'a, Sinhala/Tamil). For much of its history, these groups were not even two; they were divided into multiple competing kingdoms, occasionally allying and occasionally warring, but with religion or language playing little motivating factor overall. Colonization by European powers eliminated those states, and the British played up ethnic tensions both deliberately (in order to divide and conquer) and out of convenience (the British preferred to deal with neat ethnic groups rather than a multicultural, and thus incomprehensible, polity). The British also imported around a million Tamil laborers from their colony in India, which both connected Tamils symbolically to British rule and emphasized their perceived difference from the Sinhala population.

With decolonization, a number of insidious trends converged. First, the parliamentary democracy that followed was pretty much entirely majoritarian, meaning that a united Sinhala polity could control the affairs of the island. Secondly, Buddhist monks of the island popularized a more militant sense of nationalism based on the ancient heritage of the island in contradistinction to the Europeanized native class that tended to be more 'secular' and sometimes Christian. This left Sinhala politicians with an irresistible incentive to foment a sense of cultural militancy on the basis of exclusion of the Tamil underclass, now identified by many of the aforementioned monks as descended with various villains in the Mahavamsa (a side note: the Mahavamsa was sort of rediscovered by Euro anthropologists who likely ascribed more importance to it as a text than it possessed, but their elevation of it was definitely important in reintroducing it to Sri Lankan society). In the mid 1900s the government denied Sri Lankan citizenship to recent Tamil immigrants. In the late 60s/1970s, there was a big push for Sinhala to be the only language of state, especially teaching it exclusively in state schools and banning Tamil. Naturally, Tamil citizens in the north were outraged about being essentially excluded from the state. Protests followed in which violence on both sides led to a cycle of reprisal that was difficult to break. In response, several young militants in the north started arming themselves and demanding sovereignty for the Tamil population. After they clashed with Sinhala soldiers, they became known as the Liberation Tigers of Tamil Eelam, the LTTE, or the Tamil Tigers.

During the civil war, nationalist Buddhist monks played an essential role in motivating a policy of extermination towards the Tamil population. The monks remain the most important political institution in the country largely because of their prominence during the war. However, it's importance to emphasize that that was only a motivating factor for some Sinhala people or politicians. Many were motivated by a sense of being threatened (which they were) by "terrorists," others thought that defeating the "radical" LTTE would be the only way to help the "moderate" Tamils. Others were just motivated by hatred for Tamils expressed along a number of different markers (language, skin color, cultural practices, perceived historical slights).

What is important to note, however, is that while Buddhism, and really a particularly nationalist and particularist form of Buddhism, was a huge motivating factor for the Sinhala side, the Tamil groups were not united around any particular religion. The LTTE was officially atheistic, actually, and the Tamil-speaking people of the north of the island were a mix of a particular branch of "Hinduism" (yet another problematic category), Islam, Christianity, and atheistic. Because of the fault lines in the Tamil side over religion, the LTTE actively discouraged identification based on religion, and tended instead to focus on recent wrongs committed by the Sri Lankan government.

So the answer is probably that it was a very significant factor for the Sinhala/Sri Lankan side, although generally expressed as an undercurrent justifying multiple strands of thought, and it was fairly insignificant as motivator for the Tamil side. It's also a good example of how difficult it is to identify a distinct category called "Religion" since the roots of the conflict have much more to do with things we in the West might identify as "secular" or material, such as access to state services, linguistic or cultural patterns, etc.

Talal Asad, Genealogies of Religion: Disciplines and Reasons of Power in Christianity and Islam. The Mahavamsa: The Great Chronicle of Ceylon, trans. Wilhelm Geiger. Dan Kent, Shelter For You, Nirvana For Our Sons: Buddhist Belief and Practice in the Sri Lankan Army (highly recommended, Professor Kent was my undergrad advisor and taught a class on this very subject) Tessa J. Bartholomeusz, In Defense of Dharma: Just-war Ideology in Buddhist Sri Lanka

[might think of more sources in a bit]