Generally, it would depend on the relative danger of the area and the willingness of the other side to allow burial parties into no man's land, but there was no "no man left behind" policy like you see now.

If a section of trench was in a very active section of the front (Verdun, Ypres, etc) generally corpses were left more-or-less where they were--it wasn't worth sending more men to be killed trying to remove them. You'll read plenty of stories of people finding dead men buried in the sides of trenches, and the photographs / art of the time certainly suggest that dead bodies weren't unusual. There's a very gruesome Otto Dix painting that shows a lone german soldier sitting in a trench full of corpses--many of whom have clearly been there for a while. Let me see if I can track it down.

Here it is.

As for the dead in No Man's Land...they were left where they fell, except for when local burial truces were arranged. That was fairly rare--about the only major burial truce I can think of it was at Gallipoli in 1915, though I'm sure there were many other local truces arranged between smaller groups of soldiers.

Bodies left in No Man's Land were a fairly constant source of misery for the living, of course. In winter they froze, and in summer they bloated, turned black, and became infested with rats and maggots. In areas of heavy shelling, the bodies were basically all just pulverized and churned into the mud by the shellbursts. Unfortunately, there really isn't anything you can do about those bodies--they're just gone.

Sources

A bunch of books, pics, and art that cover stuff like this, though most only mention it in passing. All Quiet on the western front is an obvious one. I'm pretty sure there are sections about what to do with corpses in WW1 memiors like "Now it can be told" and "Goodbye to all that," though it's been a while since I read them.

Otto Dix's "Der Kreig" has a lot to say about the presence of corpses in the trenches and between the lines. Not for the faint-hearted, though--Dix can be quite gruesome.