I strongly suspect that consciousness without biology would be possible.
Imagine that you took a human being, and replaced one of his roughly 100 billion neurons with an artificial, nonbiological neuron – one that functioned the same way as the real ones, but was not alive. Would that person still be conscious? With just one of 100 billion neurons changed, I don’t see how anyone could seriously argue that he wouldn’t be.
So now let’s replace a second neuron. Still conscious? Yup. Now a third, and a fourth, and so on. Could you seriously argue that there would be a point at which too many of the neurons were non-biological and therefore the person was no longer conscious? If that’s your position, the burden is on you to explain why consciousness would cease at that point and not at any other point in the process of replacing neurons. I don’t know of anything remotely resembling a reasonable explanation for that.
If the idea of the question was to wonder how a dead body could be conscious (that is, how there could be a conscious afterlife), I think it’s pretty obvious that this doesn’t apply there, unless believers are willing to assert that after you die your neurons are replaced by nonliving functional equivalents. I suppose that there are believers who’d be willing to say something that crazy, but it’s obviously nonsense.