Is it grammatically incorrect if I use "be" instead of "are" in the following passage?

As a UK English speaker, I don't think 'be' in this context is grammatical. When you render the likes of "It's important that you be here" you are referring to the future (in which case you're using the subjunctive to describe future actions) and if you are going to use that particular phrasing, I would say it would be wrong not to use the subjunctive. However, the phrase "It is inevitable that their decisions are final" is NOT expressing suggestions, wishes or desires. And also, I would like to argue that the subjunctive mood has fallen into disuse and doesn't sound totally smooth or natural.

"Are" is better than "be".

Forms such as "It's essential that he be here" are known as the mandative subjunctive (see Wikipedia's article on the English subjunctive). They express commands or orders (sometimes recommendations): you are saying "I order that he be here." I suspect the OP is thinking of cases like this. If you're merely stating something you would use the non-subjunctive verb. "It's essential that he is here" means it's necessary, as a fact, but isn't commanding he be brought there. Likewise, "It is inevitable that their decisions are final" isn't really a command, just a statement of fact, so no subjunctive.

There are various other uses of the subjunctive, of varying degrees of popularity in contemporary English, but generally with conditionals or other forms of command ("God save the queen", etc). That doesn't apply here either.