If you include 「で」 , then the「で」will naturally be accented in speech as it is the end of the clause and the 「で」is linking the two clauses together (not really as a conjunction, but as the 連用形 for 「である」 or 「だ」).
Including 「で」 makes it sound as though the first clause is trying to explain the second. Leaving the 「で」 out makes the clause ending in 「無関係」 stand out like a separate utterance. Now the emotional force can more naturally rest on the word 「無関係」 rather than on the conjoining 「で」.
Here's another example:
Conjoined utterance with 「で」: (The two thoughts are directly linked, so the opinion expressed in the first clause sounds limited by its application to the case in the second clause.)
And without 「で」: (The first thought is now much more broad in its application and import as the speaker is essentially saying, "There is absolutely no connection between happiness and money." [full stop])
You could imagine the second sentence having a period in place of the comma (though that's not grammatically cool) to show the pause and emphasis present in speech.
In the case of 「で」 being the 連用形 of 「である」・「だ」, removing it should always make the word before it sound stronger. I can't think of any examples off the top of my head where it wouldn't.
I believe this would be unique to 「で」, but only because it is the 連用形 of 「である」・「だ」. So removing 「であり」 (in speech) would often make the last word in an utterance stronger, if for no other reason than that the speaker would basically be ending his thought there, rather than ending it on a "verb" as normally would be expected.
Just to recap the idea I was trying to get across in the example up there you could translate them differently this way:
There's no connection whatsoever between money and happiness. So, even though you get a regular job you shouldn't think that you'll necessarily be happy.
(In this case I would think that the emphasis was being placed on the application rather than the principle stated in the first clause.)
There is absolutely no connection whatsoever between money and happiness. Even though you get a regular job you shouldn't think that you'll necessarily be happy.
(In this case I would think that the emphasis was being placed on the principle stated in the first clause, the application sounding more like a secondary point supporting it)