"... that ... with ..." and "... which ... with ..."

The short answer: that can be a conjunction or a relative pronoun; only when it is a relative pronoun can it be replaced with which.

Your first example could mean either of the following:

a.) This fact, which I communicated to Mona, is irrelevant — relative pronoun.

b.) The fact that I communicated something to Mona is irrelevant — conjunction.

If we read it as a), the word that in your example has a function within the predicate of the subordinate clause that I communicated to Mona: it is the object of communicated. What did I communicate? I communicated the fact ( = the antecedent of that).

If we read it as b), that just serves to introduce the content of the subordinate clause, but it does not itself have a function within the predicate of the clause: that's why it is called a conjunction.

In your second example, however, the preposition with makes it unlikely that an object is present: you would normally not say, ?I communicated this with Mona. For that reason, we automatically read that as a conjunction here, not a relative pronoun; and only relative pronouns can be replaced with which.

OP's first sentence is only "borderline" acceptable - here's a chart showing that communicate to [someone] has largely fallen into disuse over the past couple of centuries.

If we accept that "semi-archaic" construction, note that it always requires a "secondary object" (some information conveyed - here, "the fact"). It's largely stylistic choice whether this is further qualified as being the information that was communicated, or which was communicated.

The second sentence represents the normal phrasing when there isn't a secondary object (i.e. - it's irrelevant that any communication occurred at all, and the specific information isn't mentioned). But in this case, it has to be the fact that [communication occurred]. You can't use which there.

Thus, given that which can only be used where the intended meaning is that some specific fact that was communicated is irrelevant, most native speakers will force that interpretation and overlook the "unusual" phrasing in the first sentence. Personally, I wouldn't - I'd say...

The [specific] fact[s] which I told Mona is/are irrelevant.

If you don't see a problem with to, consider "He communicated to me that I should leave". I think most people would agree this is stilted/formal/archaic.