There is no RAW for this, but with the intention of following the RAI I would look for similar effects already stated. Therefore I would rule that the effect of an Invisibility spell apply to the reflection of the vampire:
"If the recipient is a creature carrying gear, that vanishes, too. [...] Items dropped or put down by an invisible creature become visible; items picked up disappear if tucked into the clothing or pouches worn by the creature."
Within the limits of common sense, of course; for example, the vampire's reflection wouldn't become visible after attacking, making this actually a Greater Invisibility spell, and of course the effect cannot be dispelled since it is a supernatural effect.
The above description works for 3.5e; I am not familiar with the mechanics in the different D&D editions, but whatever happens to the reflection will be tied to what the game you're playing defines as being invisible.