What types of cell would need increased surface area?

It depends on the type of the cell.
Explanation:
Prokaryotic cells have internal folding of the cell membrane which are responsible for certain functions like respiration. These foldings increase the surface area.
Eukaryotic cells are compartmentalized which is a method to increase the surface area as there are lots of membranes inside the cell across which many processes take place.
In some cells like the ones making the lining of the villi in the small intestine there are folding in the outer cell membrane which also increases the surface area,