Do any heroes share romantic relationships with other heroes?

No heroes are in romantic relationships with any other heroes according to current canon. That said, several heroes have romantic relationships with non-hero characters.

• We know that Tracer is in a romantic relationship with a woman named Emily (as shown in comics)
• Widowmaker was married to a man named Gérard Lacroix (whom she killed whilst basically brainwashed)
• Torbjörn is married to a woman named Ingrid Lindholm, and his daughter is Brigitte (he also has several other children and grandchildren that are unnamed)
• Ana is married to (or has some other romantic relationship with, hasn't been confirmed they're actually married) a man named Sam.

Beyond that, no other characters have known canon romantic relationships. Some characters are related (other than friendship) in a number of ways though to other heroes and non-heroes.

• Pharah is Ana and Sam's daughter
• Brigitte is Torbjörn's daughter and Reinhardt is her godfather
• Genji and Hanzo are brothers
• Winston has an adoptive father, Harold Winston
• Orisa was created by Efi Oladele in response to an attack from Doomfist

All other known canon relations between characters are friends, neutral, or enemies. At the moment, there are no other familial relationships that have been revealed. That said, some of the friendships are strong (Tracer and Winston are very good friends, Ana and Reinhardt seem to be good friends, and Junkrat and Roadhog are close friends, for example). As are some of the rivalries (obvious ones are Doomfist v Orisa and Soldier v Reaper, though the latter two were once close associates and presumably friends).