I'm through it = I'm done

Question

Do "I'm through it" and "I'm done" have the same meaning?

Answer 1

Exactly as you've phrased the question, they do not mean the same thing.

---

I'm done.

This means that you've finished something:

It's taken me a long time to paint the house. But, finally, I'm done.

Or, idiomatically, it could mean that you're tired, in a way that's synonymous with I'm spent. Sometimes, in can be added after, or all before:

That marathon was gruelling and I need to get some rest. I'm done!
That was too much effort. I'm all done in!

---

I'm through it.

This means that you have made your way through something:

"Are you still in the tunnel?"
"No, I'm through it now.

It can also mean that you've endured something. (Which is the figurative sense of having passed through something.)

That root canal was torturous, but I'm through it now.

---

As has been suggested by comments to your question, if you add the preposition with after both through and done, then they mean the same thing:

I am through with you and your insults.
I am done with you and your insults.

Answer 2

There are four possible examples based on your question.

The first two are I'm through it and I'm through with it, which don't mean the same thing; they are used in different contexts. This is true also of I'm done and I'm done with it.

The expression I'm through can be used to signify that you have passed an exam or, if you add with, that you have stopped some employment or activity. For example:

Question: Did you pass the exam?

Answer: Yes, I'm through. I got a first.

People are more likely to say I'm through than I'm through it but the latter is possible if it refers back to the question.

Question: Are you still working in the bar at night?

Answer: No, I'm through with it/them. I didn't like the hours.

The expression I'm done can mean that I'm finished or, adding with, that I've had enough of something or brought something to an end.

Question: How's your homework going?

Answer: At last, I'm done. So I'm free this evening.

Question: I need my lawnmower back. Have you finished with it?

Answer: Yes, I'm done with it; I mowed the lawn yesterday.

Question: How are you getting along with your new girlfriend?

Answer: I'm done with her. We couldn't agree about anything.

Although these illustrations are typical of the use of these phrases, you may well find that they are also used in other contexts.