Where does the NT say the Son cannot save us unless he is God?
I'll get to that in a bit but first let's make sure we understand that we simply cannot just quote John 3:16 as though we've given all the NT states regarding salvation.
For starters, we need to ask what it means to "believe in him". Looking at the very context of John 3:16:
Joh 3:14 And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up,
Joh 3:15 that whoever believes in him may have eternal life.
Joh 3:16 "For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.
Joh 3:17 For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him.
Joh 3:18 Whoever believes in him is not condemned, but whoever does not believe is condemned already,
because he has not believed in the name of the only Son of God. (ESV)
Already we can see that believing "in him" is the same as believing "in the name of" him. So what does it mean to believe "in the name of" him? Would that not mean the same as believing in who Jesus claims to be, in who he is? Hence why the all important question: "Who do you say that I am?" Just something to think about.
We see the same in John 1:
Joh 1:12 But to all who did receive him,
who believed in his name,
he gave the right to become children of God, (ESV)
We also see that repentance is necessary:
Act 2:38 And Peter said to them, "Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. (ESV)
And quite importantly:
Rom 10:9 because,
if you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved.
Rom 10:10
For with the heart one believes and is justified, and with the mouth one confesses and is saved.
Rom 10:11 For the Scripture says, "Everyone who believes in him will not be put to shame."
Rom 10:12 For there is no distinction between Jew and Greek; for the same Lord is Lord of all, bestowing his riches on all who call on him.
Rom 10:13
For "everyone who calls on the name of the Lord will be saved." (ESV)
This is a very interesting passage. We not only see that confessing "Jesus is Lord" and believing "in your heart that God raised him from the dead" is necessary for salvation, Paul seems to tie in the confession with Joel 2:32, a passage in which states that "everyone who calls on the name of the LORD [YHWH] shall be save." This suggests that here, the confession is not merely that Jesus is Lord of one's life, one's master, but that he is YHWH.
That also ties back in with believing "in the name" of the Son. The evidence seems fairly strong that one's salvation rests, in part, in believing that Jesus is God. That's looking at it from the other end of your question regarding my previous statement that there is no salvation if Jesus is God. I will get into that when I have time.