If I learned modern Greek, would I be able to read the New Testament in its original language?

Like the others said: no. Certainly not the more educated writing, like Paul or Luke. You’d know what was going on, more or less, but you would be liable to be confused, by the syntax or by the false friends.

I’ve just gone through an exercise in Nick Nicholas’ answer to How much of a text by Aristotle or Procopius would speakers of modern Greek get?, of trying to render Aristotle and Procopius with a knowledge of Modern Greek alone. (Native Modern Greek, and I’m going to have to assume you pick up a comparable level of learnèd vocabulary.) Let me do the same with Mark and Paul.

Archaic words that might not be in your list as a Modern Greek learner, in italics. I’m outright omitting words a Modern Greek speaker would not guess. False friends followed by (!)

Mark 2:1–5

1Καὶ εἰσελθὼν πάλιν εἰς Καφαρναοὺμ δι’ ἡμερῶν ἠκούσθη ὅτι ἐν οἴκῳ ἐστίν. 2καὶ συνήχθησαν πολλοὶ ὥστε μηκέτι χωρεῖν μηδὲ τὰ πρὸς τὴν θύραν, καὶ ἐλάλει αὐτοῖς τὸν λόγον. 3καὶ ἔρχονται φέροντες πρὸς αὐτὸν παραλυτικὸν αἰρόμενον ὑπὸ τεσσάρων. 4καὶ μὴ δυνάμενοι προσενέγκαι αὐτῷ διὰ τὸν ὄχλον ἀπεστέγασαν τὴν στέγην ὅπου ἦν, καὶ ἐξορύξαντες χαλῶσι τὸν κράβαττον ὅπου ὁ παραλυτικὸς κατέκειτο. 5καὶ ἰδὼν ὁ Ἰησοῦς τὴν πίστιν αὐτῶν λέγει τῷ παραλυτικῷ, Τέκνον, ἀφίενταί σου αἱ ἁμαρτίαι

And entering again into Kapharnaum through days, it was heard that he was in a house. And many gathered, so that even the things towards the gate would no longer fit, and he was speaking the word to them. And they come bringing him a paralytic, being lifted by four. And unable to … him because of the mob, they de-roofed the roof where he was, and digging out they destroy (!) the bed where the paralytic was lying down. And seeing their faith Jesus says to the paralytic, Child, your sins are let off.

Romans 2:1–5

1Διὸ ἀναπολόγητος εἶ, ὦ ἄνθρωπε πᾶς ὁ κρίνων: ἐν ᾧ γὰρ κρίνεις τὸν ἕτερον, σεαυτὸν κατακρίνεις, τὰ γὰρ αὐτὰ πράσσεις ὁ κρίνων. 2οἴδαμεν δὲ ὅτι τὸ κρίμα τοῦ θεοῦ ἐστιν κατὰ ἀλήθειαν ἐπὶ τοὺς τὰ τοιαῦτα πράσσοντας. 3λογίζῃ δὲ τοῦτο, ὦ ἄνθρωπε ὁ κρίνων τοὺς τὰ τοιαῦτα πράσσοντας καὶ ποιῶν αὐτά, ὅτι σὺ ἐκφεύξῃ τὸ κρίμα τοῦ θεοῦ; 4ἢ τοῦ πλούτου τῆς χρηστότητος αὐτοῦ καὶ τῆς ἀνοχῆς καὶ τῆς μακροθυμίας καταφρονεῖς, ἀγνοῶν ὅτι τὸ χρηστὸν τοῦ θεοῦ εἰς μετάνοιάν σε ἄγει; 5κατὰ δὲ τὴν σκληρότητά σου καὶ ἀμετανόητον καρδίαν θησαυρίζεις σεαυτῷ ὀργὴν ἐν ἡμέρᾳ ὀργῆς καὶ ἀποκαλύψεως δικαιοκρισίας τοῦ θεοῦ,

For which you are unapologetic (!), O every human who is judging? For in what you judge the other, you condemn yourself, for you do those things, you who judge. But we … that the shame (!) of God is in truth on those who do such things. He should consider (!) this, O human judging those who do such things and do them, that you will avoid the shame (!) of God? Or do you have contempt for the richness of his usefulness (!) and his tolerance and patience, being ignorant of the fact that God’s useful thing (!) leads you to repentance? But according to you harshness and unrepentant heart, you store up treasures for yourself of rage in the day of rage and of the revelation of the just judgement of God.

The basics are there, sure (though a Modern Greek as Foreign Language learner would miss the italicised stuff). But I don’t think that counts as reading Koine, and I’m assuming less discomfort with Koine grammar than a MGFL learner would have. And those exclamation points? There’s some basic misunderstandings lurking there: loosening down the bed, not destroying it; God’s judgement, not shame.