According to Oxford dictionary, irony means
the expression of one’s meaning by using language that normally signifies the opposite, typically for humorous or emphatic effect:
‘Don’t go overboard with the gratitude,’ he rejoined with heavy irony
- a state of affairs or an event that seems deliberately contrary to what one expects and is often wryly amusing as a result:
the irony is that I thought he could help me
[count noun]: one of life’s little ironies
- (also dramatic or tragic irony) a literary technique, originally used in Greek tragedy, by which the full significance of a character’s words or actions is clear to the audience or reader although unknown to the character.
Let's consider your example sentences.
So the irony is this: technology is freeing us from technology.
When we use technology, it is reasonable (at least it seemed to appear so to whoever said that) to think that we would keep using technology. But then, how could it freeing us from itself at the same time. And that is the irony.
The postmodern reply to the modern consists of recognizing that the past, since it cannot really be destroyed, because its destruction leads to silence, must be revisited: but with irony, not innocently.
–Reflections on The Name of The Rose, by Umberto Eco.
It might be helpful to know that the context is about literature re-examination. To be specific it's about revisiting postmodern literature. Because this is written by Umberto Eco, who is an Italian semiotician, essayist, philosopher, literary critic, and novelist, we should not take his use of this irony lightly. In other word, I would not treat this use of irony the same way as used in the above example.
As Wikipedia lists 6 apparent types of irony: dramatic, cosmic, socratic, situational, verbal, and irony of fate. I believe that this use of irony in must be revisited with irony has the sense of socratic, which is used
when someone (usually a teacher) pretends to be stupid in order to show how stupid his pupils are (while at the same time the reader or audience understand the situation).
The phrase but with irony, not innocently also tells a lot. The revisiting would appear to be done innocently, but in fact not. It would rather be done in an ironic manner. Not just review it literally, but review it carefully and critically while still treating it at its face value first.
See also: http://en.wikipedia.org/wiki/Irony