Warning! Spoilers ahead for Batman: Fortress #2
It makes sense to think that if superheroes are on social media they would be friends with each other. But as it turns out, the Green Lantern Jessica Cruz has actually blocked the Flash Barry Allen on Instagram.
In terms of heroes in the DC Universe, the Flash could be considered one of the friendliest. Unlike someone like Batman, who considers his vigilantism a duty and burden he must carry, the Flash actually enjoys being the Flash. He revels in being the Scarlet Speedster of Central City. So if there is any one hero who might have a profile on social media, either as his Flash persona or as Barry Allen, it would be him. Likewise, Jessica Cruz is a more reserved hero who has to deal with her own personal issues from time to time. She enjoys her privacy, naturally so her online presence might be more private.
But in Batman: Fortress #2, by Gary Whitta and Darick Robertson, it is revealed that Jessica has actually blocked Barry on social media. When the Justice League gathers to combat a mysterious alien threat, the Flash notes that neither Superman nor Jessica Cruz is there. At this point Cyborg accuses Flash of creeping on her. The Flash defends himself by stating that asking her to be his friend on Instagram is not creeping. Then Hawkman reveals that the Flash did that after she had already blocked him.
It's never explained if Green Lantern and the Flash have accounts that are based on their hero identities or their civilian personas. But either way, it would be a bad idea for a superhero to be active on social media. If they have accounts in their hero names it would beg the question why superheroes who should be out saving people are busy taking pictures on their phones and posting them online for the world to see. Public relations, especially for active Justice League members, can be beneficial for the team. But social media probably isn't the best place to conduct that as the public would be able to see that one hero blocked another.
The Flash and Jessica Cruz having private Instagram accounts using their civilian names would also be a bad idea. If a villain learned of a hero's identity, all they would have to do is go online and see who that person is linked to in order to figure out the identities of other heroes. Because Flash is such an outgoing guy, it makes sense that he would want to befriend other heroes online. But since Jessica is more reserved, she is probably being the more responsible one by blocking him and preventing it from becoming an issue. Either way, the Flash should tone down his social media presence because Green Lantern isn't having it.