Would a profit maximizing monopolist with fixed costs but no variable costs set their price and output so as to maximize revenue (should they produce anything at all)?
Yes, if this scenario were possible the monopolist would set their output to the profit maximizing quantity. They would then charge whatever price they could get for that quantity of output. They would definitely continue to produce because they would not be below the shut-down point.
The shut-down point for a firm is the point at which its marginal revenues dip below its average variable costs. When the firm gets below that point, they are losing more by producing than by closing. However, if they can charge more for their output than their average variable costs, they come out ahead even if they are losing money. If they make more than their variable costs, they can at least pay off some of their fixed costs. If they shut down, they still have all of their fixed costs and no revenue to use to defray those costs.
In this scenario, the monopolist has no variable costs. If this is indeed the case, the monopolist will never reach the shut-down point because marginal revenue (so long as they can sell their product for an actual positive price rather than having to give it away) will never go below their average variable costs.
Therefore, in this scenario, the monopolist should stay open and produce the quantity where marginal revenue equals marginal costs. They will then charge the price that they can get for that quantity.