From a trinitarian perspective, how do you biblically reconcile Jesus dying and Jesus being God? I've heard non-trinitarians say, &quot;The Bible says God can't die, therefore Jesus isn't God.

First of all, does the Bible even say that God can't die (I assume it does, or at least it implies it). I believe that God can't be snuffed out of existence. If it does say it or imply it, in what sense is the word &quot;die&quot; or &quot;death&quot; used, and what evidence or reason do you have for your answer?

I know that Trinitarians distinguish Jesus dying (biological death of His body) and God hypothetically dying (being snuffed out of existence), but how do we prove this difference biblically? I hope that's a coherent question. To clarify, some non-trinitarians would say that distinguishing meanings of death in the Bible is just ad hoc or grasping for straws. How do we prove that this difference in meaning is biblical and not ad hoc?
The Doctrinal Position
Trinitarian position usually includes the doctrine of Jesus's having two natures in one person (Hypostatic Union).  Armed with both the doctrine of Trinity and the Hypostatic Union, the standard explanation of what happened when Jesus died on the cross is as follows:

The Divine nature of Jesus did not die or cease to exist
Neither God the Father nor God the Holy Spirit died on the cross
The body of God the Son in his human nature died and was buried
Although the human body of God the Son died, the hypostatic union of two natures was never separated, broken, or compromised

Source: Did God Die on the Cross?: The Trinity and the Crucifixion
Answering your questions
Q1: Does the Bible even say that God can't die?
Answer: Yes, see several Bible verses such as

Isa 40:28: ... The Lord is the everlasting God, the Creator of all the earth. ...
1 Tim 1:17: All honor and glory to God forever and ever! He is the eternal King, the unseen one who never dies; he alone is God. Amen.
Deut 33:27: The eternal God is your refuge, and his everlasting arms are under you. ...

Q2: [paraphrase] How do Trinitarians explain that Jesus truly died as a human being but yet retains His Divine Nature who never die?
Answer: Quote from the article above for the explanation:

As mentioned above, human nature doesn’t cease to exist in death; rather, the body perishes but the soul/spirit lives to God. Jesus’s human nature—like ours—still existed in his death, because the soul/spirit is immortal and thus the human nature still lives in/not in the presence of God. If Jesus’s human nature died/ceased to exist for three days, this would indicate not only a death of his soul, but also a split in his person—only half of Jesus would exist for three days while his body was in the tomb. We need to affirm, then, that the human soul/spirit of Jesus remained alive (thus, his nature did not die), but that he experienced a real human death like all of us: body in the ground, soul/spirit with the Lord. And his resurrected body, like ours one day, was raised imperishable and he now lives as the God-man who will never die again.

Q3: What's the Biblical evidence for the answer of Q2?
Answer:
STEP 1: Trinitarians look at the Bible verses below as the starting point (with straightforward plain sense interpretation).

Jesus is God: see supporting Bible verses such as

John 1:1,14: In the beginning the Word already existed. The Word was with God, and the Word was God. ... So the Word became human and made his home among us. He was full of unfailing love and faithfulness. And we have seen his glory, the glory of the Father’s one and only Son.
John 10:30: The Father and I are one.


Jesus died: some supporting Bible verses:

John 19:30-33: When Jesus had tasted it, he said, “It is finished!” Then he bowed his head and gave up his spirit.  It was the day of preparation, and the Jewish leaders didn’t want the bodies hanging there the next day, which was the Sabbath (and a very special Sabbath, because it was Passover week). So they asked Pilate to hasten their deaths by ordering that their legs be broken. Then their bodies could be taken down. So the soldiers came and broke the legs of the two men crucified with Jesus. But when they came to Jesus, they saw that he was already dead, so they didn’t break his legs.
Mark 15:37: Then Jesus uttered another loud cry and breathed his last.
1 Pet 3:18: ... He suffered physical death, but he was raised to life in the Spirit.



STEP 2: Trinitarians then came up with the doctrinal position above as the straightforward logical way to reconcile them, yielding the explanation in the answer to Q2 above.
Conclusion
The straightforward meaning of the Bible verses and the explanation preclude having to &quot;grasp for straws&quot;.