It really depends. Some trenches were "cleared up" by the military and civilians, and had new roads and infrastructure built over them. However, many trenches were simply left alone, either as a memorial to the war or because it was too dangerous to go near them. Many trenches still had shrapnel, shells, or even active mines still in them, and to clear them out could be fatal. To this day, there are still reports of explosions and ammunition found in parts of France and Belgium where the war was fought on the Western Front.

Here is a picture of a surviving trench from the infamous Battle of Verdun.

It is important to note much of the debris from war, much of it being unexploded ordnance, was essentially bulldozed into the trenches to fill them. There are reports of old craters being found with enormous amounts of unexploded weaponry in them. Many countries (Germany was one I remember) is actively studying old aerial images of trenches to determine where large caches of old bombs may be.

Last summer we visited a German one near Ypres in Belgium. It was forgotten then restored when a farmer discovered it decades later. It's open for tourists now, but very non descript. Surrounded by farms, no parking lot, not clearly marked.