Greetings! This is an interesting question, as OP has phrased it in a way that narrows the focus of the Empire impacting the First World War, rather than traditional impact of the war on the Empire. A challenge in terms of answering and focus, but we here at AH like a nice challenge, so let's get started on this one. As a shameless plug before we begin however, consider reading this Saturday Showcase feature I wrote on the MAIN causes of World War 1. Specifically, you might find the section on imperialism impacting the diplomatic concerns of Britain as an interesting bit of foundational knowledge.

To answer the question bluntly: indeed it did. The British Empire's sheer size at the outbreak of the First World War posed serious problems but also benefits for the leaders in Whitehall and the British armed forces. It also posed similar opportunities and threats for the governments and commanders of the Central Powers, who were all too aware before the war of what kind of threat the British Empire posed to their war plans. We can break this response down into two key "arenas of impact", strategy, and resources.

Strategy

The British Empire at the outbreak of the First World War was the largest the world had ever seen, and its territories stretched from the settler-colonies of Canada in the Americas to the "jewel in the crown" of the British Raj (modern day India and Pakistan) and the East Asian holding of Hong Kong. Should a war break out on the European continent, the planners of the War Office in London and the politicians in the Houses of Parliament feared that the colonies would also be targeted alongside the British Home Isles. In particular, they feared the security of their colonies in Africa which were neighboured by German colonies, as well as the fear of German expansion into the Persian Gulf through control (by the German Empire proper or other client states) into the territories of the flagging Ottoman Empire. The Persian Gulf, with its access to the Indian Ocean, was a key concern for British war planners who feared that the Central Powers would use it as a bridgehead to threaten the British Raj.

At the outbreak of war, the British placed their faith in a 'navalist' strategy, in which the Royal Navy (as the world's largest and most powerful) would be used to, as imperial historian John Darwin puts it,:

"Sweep the seas clear of enemy warships (and safeguard vital lanes of supply), gobble up the German colonies and impose the blockade that would steadily strangle the German economy."

The fighting on land, which the British Army was woefully under equipped to carry out (being the only army in Europe which was not maintained by a constant flow of conscripted men), would be left in theory to the Russians and French (who were far more prepared to undertake such an endeavour). This strategy quickly fell apart, despite early successes in establishing a blockade in the North Sea, defeating the Ostasiengeschwader (East Asia Squadron) of the Kaiserliche Marine (Imperial German Navy) at the Battle of the Falkland Islands in December 1914, and taking control of many former German holdings in Southwest Africa and the Pacific. German general Paul von Lettow-Vorbeck (nicknamed the Löwe von Afrika, or Lion of Africa) managed to evade British attempts to capture German East Africa however, and became the only commander to successfully invade territory of the British Empire in the entire war. In the bigger picture, the strategy failed miserably when it became clear that the main bulk of fighting would occur on land, and the failed "dash through the Dardanelles" of 1915 signalled the end of the Royal Navy's value as a strategic offensive weapon in the Great War.

Once the Ottomans joined the war, the entire British strategy from the pre-war years, which relied on the Ottomans remaining neutral or even allying with the Entente, was ruined. When this new combatant nation joined in the war on the side of the Central Powers, the British had a whole new dimension of threats to combat. John Darwin paints the picture of these threats well:

"The British now had to protect their 'veiled protectorate' in Egypt and their 'virtual protectorate' in the Persian Gulf, whose Arab statelets were superintended [controlled] by the government of India, against Ottoman attack. They had to protect the oil concession at the head of the Gulf (in Persian territory but close to the Ottoman border half-owned by the British government and intended to supply the Royal Navy. Above all, they had to blast open the Straits so that supplies could reach their vast but almost landlocked Russian ally and turn its huge manpower to account."

These concerns prompted the opening of the Middle East Front, where Anglo-French forces (aided by Arab tribes convinced to fight for the promise of an independent homeland by T.E Lawrence) would conduct battles against the Ottoman Empire until the end of the war (Constantinople did not fall, as Churchill had wished it to in 1915, but Jerusalem was captured at the end of 1917). So the British Empire's size impacted the First World War's course by quite literally factoring into the opening of new fronts, battlefields, and dragging in new parties to the conflict.

Resources

This response will not go too far in-depth on the extent to which the Empire supplied the British war effort between 1914-1918, but it will point out some key elements and sources of economic and manpower aid which enabled the British to maintain fighting strength in the Western Front and elsewhere throughout the war.

A good place (or colony to be more accurate) to start would be the contributions of the British Raj to the war effort. As soon as the British declared war in August of 1914, the British Expeditionary Force (BEF) was dispatched to continental Europe to reinforce the French defenders attempting to halt the German advance on Paris. Of this contingent of troops, a third were soldiers of the British Indian Army. Over the course of the war, a million troops from India would see action in the trenches of the Western Front, the open plains of Africa, the scorching deserts of Arabia, and staffed a large portion of the auxiliary support contingents (being cooks, manning supply depots, serving as medics, and so on). The "jewel in the crown" supplied one of the largest bodies of manpower that the British could call upon, and they paid dearly for it too (an estimated 75,000 would never see home again, and the Indian populace had to pay for the army's wages as well).

Canada also stands out as a clear contributor to the British during the First World War. It sent over an estimated 400,000 soldiers in the Canadian Expeditionary Force (which distinguished itself at battles such as Ypres, the Somme, Vimy Ridge, and Passchendaele), the largest force from any of the other dominions. Having the most developed economy of any imperial territory, Canada also contributed massively to the production of materiel crucial to the war effort. By 1917, between a quarter and a third of all British artillery munitions were being made in Canada, and it was also a key source of dollars for British war purchases from the United States. John Darwin remarks that "In manpower, industrial production and dollars, the oldest dominion had been an indispensable ally".

Hope this response sheds some light on your question, and feel free to ask some follow-ups on other dominions, territories, and impacts on the First World War as you see fit!