I'm not sure what the first video game was to refer to a way to skip levels as a warp zone, but it's become pretty common parlance. Of course, the word "warp" had already been associated with transportation because of Star Trek, but warp drive works a little differently than a warp zone. Still, they're manifestations of the same basic idea, which is abnormally fast transport.
The idea of such transportation is an old one, achieved in fantasy by means of seven-league boots and magic spells. In science fiction, however, it takes on a certain level of necessity, what with travel between solar systems being an integral part of many stories set in space. Since we don't know of any way to travel faster than the speed of light, and that would require a travel time of four years just to reach Proxima Centauri, writers of space stories have to come up with their own ways for spacecraft to circumvent the speed of light. One of the most common is hyperspace, which is basically an alternate universe where travel is faster for some reason. From what I've read, Star Trek's warp drive doesn't work exactly the same way that hyperspace normally does, instead having something to do with forming a bubble of normal space-time around the vehicle while it travels through subspace.
No, not THAT Subspace, although there are some definite similarities. The thing is, with hyperspace or warp drive, space travel is fast, but still takes time. Warp zones in video games tend to provide transportation that's instantaneous (allowing for processor delays). That's more like the idea of a magic door, or the means of five-dimensional travel used by the women in Madeleine L'Engle's A Wrinkle in Time. I often find myself thinking that teleportation would be one of the most useful magical powers to have. It would save a lot of time if I could just jump into a pipe and end up at work a few seconds later, wouldn't it? Then again, if work had a dress code, that pipe might be awfully dirty. Mario and Luigi have the sense to wear overalls when making their way through warp pipes.