Starship Troopers was clearly influential in leading to actual military research on the subject of power armor. Heinlein's detailed technical descriptions were sensible enough in terms of real engineering that they formed the basis of actual attempts to create the technology. DARPA tried (and failed) to prototype a power armor system in the 1970s, and has been working on a power armor exoskeleton (more successfully, we think?) since 2000---and clearly got many of the ideas for it from Starship Troopers. The book has had a large influence on military strategy and culture as well.
In terms of science fiction, suggestions of things like power armor have been around since at least the 1930s, but Starship Troopers was the novel that most clearly codified the concept as we know it now.
It also appears to have been a major influence in the popularity of mecha (which are sort of power armor taken to its logical extreme) starting around the 1970s, particularly with Mobile Suit Gundam in 1979.
In 1984, a SF novel called simply Armor was published as a direct response to Starship Troopers, making essentially the entire novel about the experiences of soldiers in combat with power armor. Shortly after that came Warhammer 40K, which made an entire tabletop war game around an army of warriors with power armor. Their vision of power armor is essentially the modern one (that we continue to see in Starcraft and Halo, for example), and it is quite closely based on Heinlein's concept.