I think it's better to be more explicit about pufferfish and how/why they inflate.
When they inflate underwater, then that is a ntural response to feeling threatened, an act of courtship or in some cases "practice". In the latter 2 cases, it is harmless. In the 1st instance, a pufferfish may puff up if feeling threatened or in danger, as it is a reflexive response to a dangerous situation. Fish handle stress very differently to humans (like a lot of other animals) and can die from stress. To deliberate make a pufferfish inflate is contrary to the aquarium hobby itself, it is deliberately putting the fish in danger. While seeing a pufferfish "inflate" might be the thing everyone wants to see with pufferfish, it is harmful, and unnecessary.
If puffers inflate out of air, this can be harmful. This has happened to 2 of my puffers before (neither deliberately, 1 was trimming the teeth, the other the fish had jumped out of the water) and the consequences were thankfully harmless, although this is not always the case. Expelling air from the fish can be more difficult than expelling water, in order to get them to expel the air, you may need to turn them upside down and "stroke their tummies", tail fin to mouth direction to release the trapped air. This method isn't always successful however, and can lead to an ill fish.
If a puffer appears to have "puffed up" but has a bulging stomach, rather than a round one, then that is due to overfeeding/recent feeding. An inflated puffer in the majority of cases is spherical (that is the reason for the defence mechanism, inflating spherically makes it very difficult for you to be grabbed hold of by a predators mouth), and if not, then the chances are it has been fed too much. Cutting down on the feedings, or feeding less more frequently may be a better course of action.