It's always dom. You'll never hear anyone say it as de and if you say it that way it's a giveaway that you're not from there. The course creators said they're looking for a different TTS because this one isn't really the greatest. http://www.forvo.com/word/de/#sv Here's how to pronounce it :)
I agree with Fredrik. I guess "dom" could also feel a bit too casual in some edge case, like announcing a Nobel Prize winner. Most of the time don't worry about it.
In writing it does matter though. You never use "dom" in any setting where language is important, like at work or academically. If you write "dom" while casually texting a friend they probably won't care (even though it's technically incorrect). But don't do it if your boss/teacher/colleague/customer/Duolingo is going to read it. That's the only hard rule between "de" and "dom".
You can always pronounce it "dom". Learning how to spell it on the other hand, and distinguish between the different kinds (de, dem) that's the tricky part. In fact many Swedes can't even master it... But it's not that hard to learn, it's the same as some English or American people not being able to distinguish "their" and "they're".
de is always pronounced as dom. http://www.forvo.com/word/de/#sv "dem" is the object form of "de" (just like "dig" is the object form of "du"). So for instance, if you were saying "I love you" you would say "jag älskar dig" and if you are saying "I love them" you would say "jag älskar dem" "dem" and "de" are both pronounced dom. "dem" pronounced: http://www.forvo.com/word/dem/#sv "de" pronounced: http://www.forvo.com/word/de/#sv
There are three words. De, Dem and Dom. In informal Swedish especially spoken, 'de' and 'dom' are often replaced with 'dom'. 'De' and 'Dem' are NOT interchangable and it is very very common that even native swedes can't get it correct. Read post #6 here: http://forum.wordreference.com/showthread.php?t=1855386
The correct way to spell it is 'de' but if your lucky enough Duolingo will think it was a typo and give you it right!
In old Swedish, it was all pronounced the way it was written. In Finnish Swedish, and some other dialects, it still is. But modern Sweden did away with all that because it was simpler to pronounce it all the same. The different spellings were kept however, with the result that very many Swedes nowadays can't spell these words properly. So, if you want you can pronounce it like it's spelled but you would sound quite old-fashioned in most situations.