There are very few, if any, species I outright have a problem with all-indoor exhibits for. However, most good zoo exhibits would at the very least provide natural lighting to its residents (e.g., skylights, geodesic roofs, etc.). That isn't to say there aren't benefits of outdoor exhibits that indoor exhibits can't provide, but depending on the climate or specific circumstances of an individual zoo I think it is possible to defend an indoor exhibit as the superior choice. If space and money were both infinite, however, I'd say the vast majority of species (or at the very least the vast majority of larger species) would ideally be given access to both indoor and outdoor areas, however obviously we live in the real world where zoos need to consider budgetary and spatial concerns.
As for which species are "best" for keeping all indoors, here's my first pick: naked mole rats. While super engaging to watch, they spend most of their lives underground and are very sensitive to changes in light and temperature. I honestly doubt it'd be possible to design a good outdoor exhibit for mole rats, and it's a species that a good indoor display is the best way to house them.