The phrase "while it's true that" at the start of a sentence is used to set up a contradictory idea that is to come later in the sentence. It is equivalent to "even though," "although," "in spite of the fact that," "notwithstanding," and other such words and phrases.
The basic structure is
While it's true that X, Y
where X is some thing that happened/is the case and Y is some other event or idea that one would not expect to have happened/be the case given what we already know about X.
Going through your example sentences:
While it's true that thousands of firms could go out of business if the lockdown continues, but if we re-open the economy right now there will be severe consequences.
As Weather Vane points out, in this sentence, either "while" or "but" should be removed. As the sentence is written, "while it's true that... but" is not only redundant but also ungrammatical.
While it is true that Republicans dominate the region, there is change in the air.
Weather Vane notes that the second part of the sentence does not follow from the first part, but while that's technically true in a vacuum, it likely makes sense in the wider context. The sentence probably comes from some news article that is talking about how sentiment around idea X, which is typically opposed by Republicans, is starting to shift. Gun control in the American South is a good example. We can imagine a paragraph like this:
A recent string of school shootings has caused people to start re-examining their views on gun control. While it's true that Republicans dominate the region, change is in the air.
In that case, what is implied is this:
A recent string of school shootings has caused people to start re-examining their views on gun control. While it's true that Republicans dominate the region (and so the prospect of passing gun-control legislation has traditionally been seen as unlikely given that the local government is disproportionately comprised of lawmakers who ran on a platform of opposing any such legislation), change is in the air (in regards to citizens' attitudes towards the issue, which in turn may result in their electing lawmakers with different attitudes on gun control in future elections or pressuring current office holders to change their stance on the issue).
While it is true that war has changed, it has not become less destructive.
This one is more straightforward. The sentence points out simply that:
A. War is different now than in the past (specifically how it is different depends on how far in the past we are talking, but that doesn't really matter for the sentence to make sense)
B. War is still as destructive as ever – that is, even though war has changed, it has not changed in such a way as to affect how destructive it is