Good question.

Before I get started, I think it's worth defining "trench warfare" in this context a little more clearly, because in many ways, the origins and causes of the static trench warfare we see on the Western Front had less to do with trenches and more to do with other factors.

"Trench warfare" in its broadest sense can refer to any fighting in which opposing troops face off from lines of trenches. Entrenchment and the construction of field works evolved rapidly in the late nineteenth and early twentieth centuries in response to improvements in small arms and artillery technology. We see instances of this sort of trench warfare in the American Civil War, the Russo-Ottoman War (1877-78) in the Caucasus mountains, the South African War (1899-1902), and the Russo-Japanese War (1904-05). Trenches and field works (such as breastworks or field redoubts) were a key feature in parts of these conflicts. (See Murray, the Rocky Road; Strachan, European Armies; Howard, "Men Against Fire")

All belligerents went to war in 1914 prepared to use trenches, and field works appeared everywhere, just as they did in the conflicts leading up to the First World War. What made the Western Front unique is that the front line was static. To describe this type of fighting, I prefer the German term "Stellungskrieg" (meaning a war of position or positional warfare) because it emphasizes the static nature of the fighting. Trench warfare was an important component of Stellungskrieg, but trenches were not the root cause. (Strachan, The First World War vol. 1; Dykstra, "Entrenchment, field fortifications, and the learning process in the BEF")

To answer your question in brief, there were instances of Stellungskrieg outside of the Western Front. Obviously, they were nowhere near as widespread in terms of time and space, but it happened.

Before I dive in, I think it's worth clarifying the causes of Stellungskrieg.

Causes of Stellungskrieg on the Western Front

Like I noted above, there were plenty of instances of trench warfare before the Stellungskrieg of the Western Front. There are two main reasons why the Western Front stabilized in 1914. The lack of an effective mobile arm applies here, like it did everywhere else, but the Western Front was unique in two ways:

1. Unprecedented concentration of forces per mile: On the Western Front, there were more rifles, machine guns, and artillery pieces per square mile than anywhere else (by a long shot). Busting through this quantity of materiel was enormously costly.

2. Highly developed rail networks: If you managed to bust through your enemy's defenses, he had access to developed rail networks to redeploy forces and close gaps. This is why, in fall 1914, neither the Allies nor the Germans were able to outflank the other after the Battle of the Marne. It's also why the Germans had a tough time cracking the British Army in 1918.

The first is the most important I think, because where we have tight frontages with lots of men, we get instances of Stellungskrieg in the First World War. (See Edmonds, Military Operations 1914; Strachan, The First World War vol. 1)

Now that I've gotten this excessive background out of the way, I'll analyze the major fronts and describe any cases of trench warfare or Stellungskrieg.

The Eastern Front

No true Stellungskrieg appeared in the East because there was a relatively low concentration of men and materiel per mile of front and because the Russian rail networks were comparatively sparse and underdeveloped. Trenches were dug, but if an army couldn't break through with a frontal assault, it was always possible to simply maneuver around a flank. The Battle of Tannenberg in 1914 is an excellent example of how mobile war persisted in the East. Later offensives in 1915-17 all demonstrate that the front remained fluid. (The best book on the East remains Stone, The Eastern Front, in which the nature of the mobile fighting is an emphasis)

Italy

Along the Isonzo river in northern Italy, we have an authentic instance of Stellungskrieg. The front was quite narrow (400 or so miles) and the concentration of forces was pretty high. When the Italians first attacked the Austo-Hungarian defenders along the river in 1915, they failed to break through. Both sides dug in as best they could given the difficult terrain, and the line hardly moved until 1917. There were some 11 battles on the Isonzo before the Germans finally cracked the Italian positions at the 12th battle in 1917. Trench networks here were never as advanced as the Western Front, mostly due to the mountainous terrain, but the fighting was position warfare in the truest sense. (Most of what I know comes from Herwig, The First World War, but there are full volumes on the Isonzo that will have more detailed information about the tactical nature of the fighting)

Africa

Not much to be said about Africa. The frontages were simply too large and the concentration of forces too small to produce Stellungskrieg. (See Strachan, The First World War vol. 1)

Middle East

The Gallipoli peninsula hosted some pretty static fighting in 1915. Again, the length of the frontage here was tiny (only 12 or so miles across), making the concentration of forces pretty high (couple hundred thousand men so men on either side). Like on the Isonzo, both sides dug in, but the soil was hard and difficult to work with and the terrain was extremely mountainous. In addition, British forces were only there between March and November 1915, so there wasn't much time to develop the sophisticated defenses that we see on the Western Front. (For Gallipoli, see Prior, Gallipoli: The End of the Myth)

An example of trench warfare without Stellungskrieg*

Like I noted above, "trench warfare" did not necessarily mean Stellungskrieg. Consider the Siege of Kut as an example. After British forces took the city (south of Baghdad) from the Turks in 1915, they were left without much support. They hunkered down in the city and were besieged by Ottoman forces between December 1915 and April 1916. The Turks used trenches to create approaches to the city, and British attempts to relieve the city involved a good deal of fighting from trenches. This was not true Stellungskrieg, however, as the city eventually fell. It was siege-like, obviously, but there was still movement in the sense that the trench networks collapsed (both in April 1916 when the Turks took the city and when the British captured it again later in the war). (Rogan, The Fall of the Ottomans, an excellent book)

Analysis and Conclusion

To summarize, static trench warfare (Stellungskrieg) emerged as the product of high force concentrations per mile of frontage. Without an effective mobile arm to pierce defenses and penetrate deep beyond lines, static fighting was the result. The true Stellungskrieg that we see in the West was never totally replicated, but the fighting on the Isonzo and Gallipolli came close. The former was very similar to the West in that there were bloody frontal assaults that returned almost no gain in ground. The difficult terrain, though, ensured that trench networks lacked the sophistication of those in the West. There was no Stellugskrieg on the Eastern Front because the concentration of forces was way lower (fewer men over a much larger area). I've also argued that you can have "trench warfare" without Stellungskrieg (the wars before WWI and the Siege of Kut as my examples).