In Trigger editor, try File > Export triggers. It generates .wtg and .wct files that you can import to your map.
Be careful though, importing triggers overwrites existing triggers.
But this doesn't do the same thing as copying triggers as text. You can open .wtg files with a text editor, but you can't read them like you would with "copy as text".
You can also extract the file war3map.j (with WinMPQ for example), I think it contains all the triggers, it's jass code though.