Will the monopolist inevitably increase the price if demand grows?
In this post, we find that the optimal monopoly price can be written as

$$P^* =  \frac {|\eta|}{|\eta|-1} MC $$

where $\eta$ is the elasticity of demand with respect to price, and so optimal price will necessarily be at a point where $\eta$ is higher than unity in absolute terms.

From this one can deduce some unambiguous and some ambiguous scenarios when the demand schedule shifts outwards, e.g.

SCENARIO A: constant price elasticity of demand, convex costs: $\implies$ Price and quantity will increase.

SCENARIO B: constant price elasticity of demand, linear costs: $\implies$ Price will stay put, quantity will increase.

The floor is yours. 

PS: Note that to have constant price elasticity of demand, the demand function will necessarily be non-linear.