I think it would be possible, if cumbersome, for a hypothetical language to avoid proper nouns. Writing the aliens' dialogue in a book might not be fun, but it might work for an occasional detail.
The question is, what function do proper nouns fill? And can anything else do that job?
Proper nouns pick out a particular entity or set of entities. They are a shorthand. There are other ways to do that, often used if you don't know the name for a thing. These include possessives, predicates, and pronouns.*
Case: You have a dog with no name.
Solution: I say "Your dog." The possessive + a singular noun narrows it down to a single entity.
Case: But you've had other dogs in the past. You are unsure which "your dog" I mean.
Solution: I say "The dog you have now."
Case: You have a few children, one of whom I want to refer to.
Solution: I say "Your second child." Or perhaps she's the only girl. I say "Your daughter."
Case: There are photos of my cousins in a photo album.
Solution: I say "Have you ever met the one who's wearing a white shirt in this photo?"
Case: You find it hard to identify which cousin I mean.
Solution: I say "This one" and I point.
Case: I want to discuss the Jesuits.
Solution: I say "The missionaries who started going to other countries about four hundred years ago and who have been to all six continents."
Notice that the last one requires a few qualifying predicates; I can't even say Catholic or identify a single country they went to, at least by name. And it might still not be enough for you to realize that I mean the Jesuits. Even worse, we may not know the same things about them. You might think I mean some other set of missionaries because you don't happen to know how many continents the Jesuits have been to or in which century the society was founded.
Moreover, we might not even agree on a predicate. You and I could talk about "the greatest English writer" and I could be talking about Shakespeare when you mean Ian McEwan.
Those are the cases that would be harder to write.
Incidentally, even proper nouns aren't always enough to do this job. If you address "Luke" now, we know you mean me. But if there are two people named Luke in the room, you have to say "Luke S." or "Luke A." But what if a person's full name isn't unique? Say there are two people named Luke Sawczak. Now you have to fall back to the same strategies shown above: "The Luke Sawczak who was born in Canada" or "That one there"... and in sentences like those, the name looks awfully like a common noun, not a proper noun.
* I mention that pronouns are one of the strategies to avoid proper nouns. Obviously these are a huge help. And I can't imagine a language that could function without them. Even to identify the speaker ("I") or the listener ("you") in any conversation would become monumentally difficult. That said, there are languages that have bigger sets of pronouns than we do whose speakers might wonder how on earth we get by... :)
