Small arms used in WW1 were for the most part the result of doctrinal inertia and decisions made in the decades leading up to the war. While there existed rifles in 1914 that were roughly on par with the bolt-action service rifles of the day (like the Winchester 1895 and Savage 1895), technology and doctrine prior to the war led to the widespread adoption of bolt-actions.
Going all the way back to the earliest available lever-actions, the problem was a reluctance to adopt magazine rifles in general. At a time when supply was a major issue for even armies relying on slow-firing muzzle loading rifles, magazine rifles faced resistance from leadership that feared undisciplined soldiers would fire off ammunition in a panic and exacerbate already poor supply situations. Thus, armies across the world moved towards single-shot breech-loading systems instead of magazine rifles like breech-loaders. During this period, bolt-actions became popular, although outliers like the Trapdoor Springfield, Martini-Henry, and Werndl stand out. Gradually armies became less averse to magazine rifles and began experimenting with such, but the available magazine rifles at that point generally drew heavily from existing systems, and, in the case of the Gewehr 71/84, were modifications of existing service rifles.
By the time armies had started considering magazine rifles, lever-actions had failed to gain widespread adoption for another critical failing. While service rifles of the time were expected to have an effective range of several hundred yards, lever-action rifles of the time were chambered in weaker cartridges that had significantly shorter effective ranges. Instances exist dating back to the Civil War of piecemeal adoption of lever-actions with great effect, but the short effective range of these weapons was generally noted in those instances. It wouldn't be until the development of the Winchester 1886 that there existed a widely-available lever-action rifle chambered in a standard military rifle caliber of the time (.45-70), but that very same year, smokeless gunpowder caused another radical change in rifle development.
The shift to smokeless powder would lead to the development and adoption of the rifles used in WW1, and with it came a new generation of significantly more powerful cartridges. The added power of smokeless powder cartridges would force the development of entirely new weapons, as existing black-powder rifles weren't capable of handing the kinds of pressures produced by the new ammunition. Here was the first real opportunity for a lever-action rifle to be adopted as a standard service weapon. Unfortunately, I'm not familiar enough to speak in detail about the major powers, but the end result was widespread adoption of more bolt-action rifles. The French hastily reworked the Kropatschek into their 1886 Lebel, the Austrians adapted an update of the 1886 Mannlicher (Steyr 1888), the Germans developed the 1888 commission rifle that was a mix of several different designs. Most other powers adopted something that was either a development of a Mauser or Mannlicher system. The real outliers were the British and Americans, but as far as I know the British were the only power to consider anything resembling a lever-action rifle for their magazine rifle.
When World War 1 broke out, every power went in with either a variant of their original smokeless powder rifle or a similar bolt-action system. The extreme demands of the war would cause a surge in rifle production, but production concentrated on what was already in service. The only instances of lever-action rifles that were adopted in frontline use were several hundred thousand Winchester 1895s purchased by the Russians owing to extreme rifle shortages. In secondary roles, we see some older Winchester models adopted by the French and British, and a small order of Savage Model 1895s purchased by Quebec to free up standard weapons for the front. One thing to note is that the Winchester 1895 did perform reasonably well and would see extensive service as late as the Spanish Civil War.
In theory a lever action rifle might seem to be just the thing for combat in the early 20th century, since the large size of the magazine (more than a dozen rounds often) would be a huge advantage. While such rifles were used in WWI, they weren't ever adopted as the main service rifle for any major power. When you look at the tradeoffs in industrialized massed warfare a lot of the seeming advantages of a lever action rifle don't hold up.
Lever action rifles use tubular magazines, while bolt action rifles use vertically stacked magazines. One of the concerns with tubular magazines is proper feeding with rounds that have pointed projectiles, it's potentially easy for the point of a round to get pushed to the side in a tubular magazine, where it might get wedged beside the next round. Also, any jam in a tubular magazine is harder to clear compared to a box style vertically stacked magazine.
Most turn of the 20th century lever action rifles use ammunition that for the time was impressive but became less so as rifle technology advanced. Most such rounds are basically upscaled pistol ammunition, with lead "ball" projectiles (a classic example being the .44-40 Winchester round). This was great in 1870, but by the 1880s bolt action rifles were using ammunition with twice the muzzle velocity and copper jacketed aerodynamic bullets (FMJ, spitzer, or expanding projectiles). Twice the muzzle velocity translates to 4x the muzzle energy. Better aerodynamics and higher speeds means much better accuracy and much longer ranges with superior terminal ballistics. These rounds have much longer overall lengths than those old lever action rounds, so if you wanted to build a lever action rifle that shot them you'd get correspondingly less magazine capacity (since the capacity is more or less equal to the functional magazine length divided by the individual round length). Which meant that you had to take some major compromises if you wanted to use lever action rifles instead of bolt action rifles. You had to accept using lower powered bullets with shorter effective ranges, less accuracy, and weaker terminal ballistics, or you had to accept lower magazine capacity.
Most of the militaries of WWI weren't willing to compromise on bullet range and accuracy, which is a major reason why bolt action rifles were so ubiquitous. Additionally, many armies used machine guns that fired the same rounds as the service bolt action rifles. For example, the Chauchat and the Berthier/Lebel, the Mauser and the MG 08/15, the Lewis Gun and the Lee-Enfield. Using common ammunition for service rifles and machine guns is a huge logistical advantage because you don't need to split up manufacturing capacity, have separate sets of tooling, etc.
An even more important factor here is reloading. If you're in a firefight that lasts 2 minutes a Winchester lever action is fantastic, 15 rounds is a tremendous amount of ammunition in small engagements, and you will likely not need to reload. However, if you're in a battle lasting hours or even days, with massed forces on both sides, reloading is going to be a much bigger factor. And here the lever action shows its great weakness. A typical lever action must be reloaded one round at a time through the loading gate. A typical WWI-era bolt action rifle (such as the Berthier, Mauser, or Springfield) could be reloaded from a clip (5-rounds being common). With a bolt action rifle you do have a shorter amount of firing time and you have to reload more often, but being able to reload 5-rounds at a time quickly means that there is less downtime and the time averaged sustained rate of fire is higher. Which, in the context of shooting as part of a multi-person fire team or squad means that the overall volume of fire is higher and the windows of vulnerability during reloading are shorter and less likely to overlap between individuals. Also, it requires less manual dexterity to push a clip into a magazine than to individually load 5 rounds through a loading gate, this is important in battlefield conditions.
Edit: I can't believe I forgot to mention one of the most important factors: shooting prone. When shooting prone or shooting from a rest (such as a berm or the side of a trench) with a bolt action rifle you don't necessarily even break your sight picture, and you can keep your overall body position. When shooting prone with a lever action rifle you have to raise the gun or turn it sideways, break your sight picture, break your body position, etc. All of which is undesirable if you're trying to place well-aimed shots at long distance.
More can always be said but /u/Meesus covers the topic here.
/u/BillyDeeWilliams1990 covers it here as well.
While the answers so far have been great, one thing to add is that one nation did use lever action rifles in the first world war. That nation being Imperial Russia, who in addition to their Mosin Nagant rifles used Winchester M1895's chambered in 7.62×54mmR (R denoting rimmed). Which used stripper clip fed, fixed box magazines rather than tube magazines. Which when coupled with a new locking system allowed for the use of spitzer cartridges (such as 7.62×54mmR). In total the Russias ordered 300,000 of these rifles which were used both on the eastern front against the Germans and also during the following Russian civil war.