How difficult would it be for a person who knows Koine Greek to communicate with a native speaker of modern Greek, aside from the non-existence of words for modern inventions?
It's anecdotal, but whenever I taught ancient Greek, my modern Greek students were usually the first to drop. It is not at all what they expected, and they were not happy about the ancient pronunciation. Koine pronunciation is closer to Modern Greek than Ancient Greek, although it should be pointed out that there's no one way Koine was pronounced. It differed not only regionally but also chronologically. The same is true, of course, for ancient Greek, which is why textbooks usually teach Attic Greek and then only cover the dialects later.
Beyond pronunciation, there's also a vocabulary difference. The modern Greek word for dog, e.g., is σκύλος, while in ancient Greek it was "animal hide." You also have plenty of modern words (not necessarily inventions) that were borrowed from other languages, especially Slavic and Turkish, that would appear quite bizarre to a Koine speaker.
But the largest difference is in grammar. Modern Greek lost infinitives, optatives, participles, and duals; merged the dative and genitive cases; gained gerunds; has some differences in conjugation endings; and uses more periphrastic verb forms (such as using θα να for future constructions; also, να is a modern particle that would throw ancients for a loop). I have my doubts that an ancient Koine speaker would know what to do with είχα γράψει or θα να, but they would at least recognize that the former had something to do with writing.
All of these would have rendered the two languages somewhat unintelligible, getting, say, the mere gist of the languages rather than any specifics, even if they were to pronounce everything the same.
Quite difficult. The pronunciation has changed significantly from Koine to Modern Greek, and anecdotally, my Modern-Greek-speaking friends and I usually have to write out words when discussing them: even when the same word exists in both time periods, I can't understand their pronunciation and they can't understand mine.
In writing, on the other hand, we can usually recognize a lot of the words. The grammar has changed significantly, but there's enough similarity to get the basic meaning across…if they use a form of the alphabet that both can understand. All-caps (imitating old Greek inscriptions) is the most likely to work for this, since Koine handwriting and Modern handwriting are significantly different from each other.
@Draconis pointed out that Homer is apparently a particularly hard to read example, and instead pointed me to what I believe is an example of Koine Greek (here) which I find relatively easy to read. To the extent where I wouldn't even have been sure if this is Ancient Greek or just more archaic Modern Greek. If that is the kind of Ancient Greek you mean, then yes, that is indeed mutually intelligible with Modern Greek.
Then, to keep things interesting, @cmw linked me to this which is somewhere in the middle. I can read that more easily than the Homeric example above, but not so easily that I would call this intelligible. At best, I can get the general idea. I can actually understand the French translation more easily and my French isn't that good.
In other words, it seems like the answer to your question will depend entirely on exactly which version of Koine you are using. Some will be mutually intelligible with Modern Greek, others won't be. At least not to this Greek.
About as close as Latin and Italian as far as I can tell. As a native modern Greek speaker, I can read ancient texts in that I know the alphabet, but I would usually understand no more than the general gist. I certainly couldn't communicate with someone speaking or writing Ancient Greek. For example, I just tried reading the first stanza of the Odyssey:
ἄνδρα μοι ἔννεπε, μοῦσα, πολύτροπον, ὃς μάλα πολλὰ πλάγχθη, ἐπεὶ Τροίης ἱερὸν πτολίεθρον ἔπερσεν: πολλῶν δ᾽ ἀνθρώπων ἴδεν ἄστεα καὶ νόον ἔγνω, πολλὰ δ᾽ ὅ γ᾽ ἐν πόντῳ πάθεν ἄλγεα ὃν κατὰ θυμόν, ἀρνύμενος ἥν τε ψυχὴν καὶ νόστον ἑταίρων. ἀλλ᾽ οὐδ᾽ ὣς ἑτάρους ἐρρύσατο, ἱέμενός περ: αὐτῶν γὰρ σφετέρῃσιν ἀτασθαλίῃσιν ὄλοντο, νήπιοι, οἳ κατὰ βοῦς Ὑπερίονος Ἠελίοιο ἤσθιον: αὐτὰρ ὁ τοῖσιν ἀφείλετο νόστιμον ἦμαρ. τῶν ἁμόθεν γε, θεά, θύγατερ Διός, εἰπὲ καὶ ἡμῖν.
I recognize various words (e.g. ἄνδρα, πολύτροπον, πολλὰ, ὣς etc.) but I can't get any meaning out of this. And I actually know what it is saying! Maybe if I spend a long time trying to decipher it I could come up with some sort of approximation, but frankly, it would be no easier than reading Latin and trying to decipher that using my Spanish, Catalan and French. In fact, I suspect the Latin will be easier simply because I speak three Latin languages so would have more to draw on.
Now, I am not a particularly erudite Greek speaker nor do I have an especially large vocabulary. Although I was raised in Greece speaking Greek, my academic and professional careers have been in other languages. I am sure that a more erudite modern Greek speaker would understand a little bit more, but I cannot believe anyone who doesn't actually know Ancient Greek would understand this.
So, to answer your main question, Ancient and Modern Greek are not mutually intelligible. Barely at all.
I actually am greek (and my mother language is greek and I've lived in Greece all my life) and was taught ancient greek for some years in high school. The conclusion? It was like a foreign language.
Ancient greek at that time seemed more difficult to me than english or french (because it had some crazy difficult grammar). It still is, I can't understand anything from ancient greek. I do know the meaning of some ancient greek phrases by heart but it would be the same as knowing chinese phrases. The fact that some (few) words are common doesn't really help.