It would seem that there is no mention of "hell" in the modern sense occurring in the Bible.

Correct. The idea of a Satan/Lucier/Beast/Adversary tormentor ruling over the fire-and-brimstone hellscape is unsupported by Biblical text.

I wonder if the modern Church came to twist Jesus' mention of Gehenna or the Lake of Fire in Revelations to mean some kind of eternal punishment, when in fact that was not the case.

Depends on what you mean by "modern Church." Dante had a massive impact on our modern conceptions of Hell.

"Dante and Shakespeare divide the modern world between them; there is no third. But even more than the theater of Shakespeare, Dante’s poetry was the compass according to which the direction of the modern world could be determined, both poetically and existentially." TS Eliot, 1929

Source supporting Dante's impact: https://dlcl.stanford.edu/sites/default/files/files_upload/lummus-inferno-influence_1.pdf, "Dante's Inferno: Critical Reception and Influence," D Lummus.

Dante is the driving force behind

• beliefs in the existence of a modern Hell
• Lucifer as a figure in Hell, Lucifer as ruler of Hell
• demigod tormentors who torment those imprisoned in Hell
• the fire-brimstone linkage with Hell, and the physical landscape of Hell
• the dissemination of the belief that judgement of individual souls will precede the Resurrection. Speculating here, I assume this is the product of commoners visualizing Dante's writings and mistakenly imagining the realms of Hell and Heaven existing contemporaneously to themselves, which necessitates judgement at death rather than at Resurrection/Rapture. Biblical passages (Deut 31:16, Job 7:21, Job 14:12-14) state that dead souls will rest... until Resurrection or Rapture, depending upon interpretation.

Source supporting my delayed judgement discussion: http://people.umass.edu/lrb/files/bak05deaS.pdf, "Death and the Afterlife," LR Baker.

As far as I know, Dante is not the source of

• the conflation of Satan, Lucifer, the Beast, and the Adversary
• belief in the existence of demons, daemons, or fallen angels who are agents of the ruler of Hell
• the popularization of the word "Hell" rather than Hades, Avernus, or Gehenna. Some of the Anglosphere's consolidation of conceptions of the netherworld around the world "Hell" is definitely attributable to popularity of translations of Dante's Comedy though

Might want to try /r/AcademicBiblical

Hi, there was a recent question on Hell; check it out: the thread contains links to a bunch more

• When did Hell become a torture furnace?

Jesus actually does speak about a place of fire and eternal punishment, so that part is actually from the inception of Christianity (or at the very least as old as the gospels).

If your hand or your foot causes you to stumble, cut it off and throw it away. It is better for you to enter life maimed or crippled than to have two hands or two feet and be thrown into eternal fire. ([Matt 18:8] (https://www.biblegateway.com/passage/?search=Matthew+18%3A8&version=NIV))

Matt 25:46 makes it clear that the damnation is eternal:

“Then they will go away to eternal punishment, but the righteous to eternal life.”

So the concept you mention is clearly present in the Gospels. As for the "modern sense" of Hell, there are quite a few different depending on the denomination. For example many theologicans believe that Jesus was speaking metaphorically and Hell is not an actual physical place of fire but rather a mental state of exclusion from God. But there is no basis for this interpretation in the Gosples.