If I change this sentence
We could not communicate through the phone.
Through the phone, we could not communicate.
Does it still remain grammatically correct? Is it OK like that? What's the difference?
Both of these are acceptable, though the focus in each is different.
By placing different parts of sentence at the front, you make them more important.
In the first sentence, the focus is on communication and the inability to do it. You could extend this as follows:
We could not communicate through the phone, though we could use it to (do something else).
In the second, the focus is on the phone and its uselessness in the situation. It could be extended thus:
Through the phone, we could not communicate but through the (something else), we could.
This works when written. However, in spoken language, stress and intonation could alter the focus regardless of word order.
As a matter of preference, I would tend toward the first example, simply because it is the natural order of the sentence without clauses being juggled and extra punctuation being added. I would only choose the second if I specifically wanted to alter the focus, as explained above.
Hope that helps.
Actually, I get the feeling that you are strugging with the arrangement of this sentence because of the preposition "through." Communication is what takes place between two or more people, and it does not take place "through" a phone. Perhaps something like "While speaking on the phone, we could not communicate well," would suffice (or "we could not communicate while speaking on the phone."