In the generic sense, you could use edit, since, strictly speaking, removing is a form of editing. That would be much better than behead.
You might also consider the verb dele, though. It means:
To remove, especially from printed or written matter; delete.
That word itself won't tell you if the text to be deleted comes from the front or the back of written work, but you could specify that by saying, "dele from the front," or, "dele from the end."
As Bill suggested, though, the words remove and delete would probably sound more natural. If any English words do mean "delete from the beginning" or "delete from the end of something," they don't spring to my mind.
If you wanted to use something informal, you might consider lop off or chop off. One dictionary defines lop as:
To cut off as the top or extreme part of anything
so lop off might at least carry the implication that the text being deleted is from either the beginning or the end, and not the middle, whereas that might not be so for remove or delete.