I can't really think of what tactical value that would have.

If you had a full-scale assault in mind, you'd need a ton of such passageways to funnel even a single division's worth of troops into enemy lines. This kind of project would not only alert the enemy as to your plans for an attack, but the defenders would know where these passageways were (thanks to aerial reconnaissance and seeing a shovel or pick come through the wall of your trench). In the First World War, we see a general trend toward thinner skirmishing lines and a dispersal of men on the battlefield. Stuffing an attacking force into the narrow confines of a passageway like that would result in catastrophic casualties. Simply training a machine gun or two and some artillery on it would spell the end of the assault.

The other possible use for such a passageway would be trench raids. Beginning in late 1914, when the front stabilized and both sides settled in for the winter, armies began launching raids to foster the "offensive spirit" of the troops and learn information about opposing forces. Key to raids was the element of surprise. They usually happened under cover of darkness and were carried out by just a few men. Such a passageway might be handy to get men safely across no man's land, but again, the defenders would be able to see where this passageway was, nullifying the advantage of surprise so important to a trench raid.

The general idea of pushing trenches as close to the enemy line as possible, but not actually into the enemy trenches, however, was a common tactic. Such a trench that extended forward and perpendicular to the front-line firing trench was called a sap. It had several uses.

First, saps could serve as observation posts. A sentry would occupy the sap and keep an eye on things. If the sentry troops spotted trouble, they'd run back through the sap and report back to the firing trench.

Second, saps were used as "bombing trenches." That is, troops armed with grenades would advance up the sap and get close enough to the enemy trenches to toss in grenades or other explosives.

Third, the Germans pushed their Minenwerfer (literally "mine throwers," basically a mortar) into saps to get them close to the enemy's trenches, where they'd be easier to aim and find targets.

Armies also dug under the enemy's trenches and planted mines beneath them. I'm not going to jump into a ton of detail here, but if you are interested in more, see my source list below. The most dramatic example of mining was the battle of Messines in 1917. British miners deposited about 450 tonnes of explosives beneath German Fourth Army. One of my favorite First World War quotes concerns the mines: General Plumber, the commander of British Second Army, remarked on the evening before the battle (June 6), "Gentlemen, we may not make history tomorrow, but we shall certainly change the geography." The resulting crater was 250 feet wide and 40 feet deep. It successfully blew the German trenches into oblivion but was also next to impossible to advance over.

So the TLDR of my answer is that digging a passageway straight into an enemy's trench line wouldn't be of much use. Getting closer to opposing trench lines was tactically valuable, however, and armies regularly mined beneath the enemy's trenches to plant explosives.

Sources: Various records from the WO 95 series at The National Archives; Craster, Fifteen Rounds a Minute; Barton, Beneath Flanders Fields (not the most academic of books, but still decent); Edmonds, Military Operations: France and Belgium 1917

Trench warfare tactics did evolve over time.

At the time of the Battle of the Somme in 1916 it was actually the practice to dig a 'sap' toward the enemy trench as a jumping-off point. Initially, in most places there was a reasonable distance between the front-line trench positions on the Somme, because of the ground, and this practice was a way to get closer under protection. Attacks toward Thiepval were carried out in this way. In other areas, troops rose from a trench parallel to the trench being attacked. This had the advantage that the enemy's fire was spread: attacking from a forward-cut trench meant being effectively 'in enfilade' when attacking.

As the Somme battle wore on, parts of trenches were taken and then incorporated into the line, a situation that caused considerable confusion. The subsequent micro-management of attacks, entire battalions being used to take a particular section of trench, and nobody knowing where they were anyway because the trench system was smashed to smithereens, were part of the tragic aspect of this battle.

During 1917 and 1918 tactics evolved. The practice of driving a sap forward was generally stopped: such a position was obvious and vulnerable. The sides generally kept their lines parallel, the ground and height advantage dictating positions. The key to taking trenches came to focus on using artillery to destroy, or at least shock, the enemy front positions. This was actually very successful if very tight coordination was achieved, and during 1917 many generals became very confident they could take a German trench under almost any conditions. The espousing of this confidence, and the subsequent poor results, were a key reason parts of the French Army mutinied in April 1917.

For the British (which includes the Empire troops used in these battles), such tactics worked well at Messines and Vimy Ridge. A nice, parallel approach was ideal for artillery planning, of course. The problem was this only worked well on a front-line trench: targeting and timing became impossible for the second line of defence, and the one after that, and the artillery was also quickly out of range. In these attacks, troops often formed up in no-man's land immediately before the attack, in the pre-dawn darkness.

The Germans realised this, and changed their defensive approach to have a lightly-manned outpost line, then a strongly-held main line, and the Hindenburg Line to which they retreated in early 1917 was built on this idea.

After March 1918 most of the fighting took place outside the old battlefields, the war finally became more mobile. Intricate trench systems disappeared, there was not the time or manpower to build them.