So, my question is this: If I were to create a wall border made of christmas lights and plastic vine, would the vine melt and/or catch fire?

Depending on what it was made of, it could. They sell long Christmas light boughs of evergreen (fake) during the holidays..

Its made of plastic...

Yeah but some plastic is different than other plastic, made to be flame retardant and heat resistant, like the plastics they use for children's toys, while others are not meant to be used in heat or by flame at all. Do you have the package for this stuff still? You can look up the 'conforms to" information to find out about whether it's flame retardant or whatnot.

I haven't bought it yet. I would just purchase some plastic rose vines from Walmart... do you think that stuff would be okay next to white christmas lights?

They make bulbs that have a big piece of plastic that surrounds the bulb, see if you can get some of those. That way you won't have a hot bulb right on the plastic vines. If you do decide to do this, turn it off before you leave or go to bed of course.