Dabi and Tomura were very disconnected during their first meeting. Dabi immediately insulted Tomura and refused to tell him his true name. Dabi also doubted the League of Villains' allegiance to Stain's ideology. This eventually caused Tomura to try and attack Dabi who was ready to defend himself but Kurogiri stopped it.
Even so, they became comrades after Tomura found his conviction following their first meeting. Dabi follows Tomura's orders and he trusts in Tomura taking charge of the League of Villains. Following All For One's defeat, Dabi becomes a valuable ally and partner to Tomura.
He also appears to respect him, as he was amazed by the latter’s power in the Meta Liberation Army Arc and was pleased he gained an army from their victory.
Dabi appears to view Himiko as a valuable ally. Even so, he doesn't appear to have a high opinion of her personality and often refers to her as crazy. In their first meeting, Dabi doubted the League of Villains' motives just because of Himiko's desire to join them. They do, however, become comrades. Since they were both admirers of Stain, they shared a common goal in wanting to bring down the hero society.
Dabi does not tolerate Himiko making fun of him when things do not go as he would like, as when she accuses him of not having social skills and burning those he does not consider fit to be part of the league, or being salty because his fire had no effect on Gigantomachia.  Generally, Dabi responds to these provocations with blunt sarcasm.
Dabi is shown to have no problem working with Twice and is seemingly not annoyed by his antics and contradictory personality. In return, Twice has confidence in Dabi's strength and rebuffs his doubt about his own strength. It is implied that Dabi and Twice view each other as brothers in combat. 
However, there may be occasional arguments between the two, given that Dabi stated he does not care what happens to his fellow League members, which conflicts with Twice's strong sense of camaraderie. When Giran was kidnapped by the Meta Liberation Army, Dabi does not hesitate to consider it nonsense to go and rescue him, which generates a violent response from Twice.
Dabi and Spinner share similar interests in taking down the hero society as both are huge followers of Stain's ideology. However, Dabi seems to disregard Spinner's presence like with most of his fellow League members. He even complains about Spinner's driving skills to the point the latter would shout back with annoyance. 
Dabi often snarks at Spinner's expense and insults him whenever the opportunity comes by. When Spinner later reveals his motivations for joining the League of Villains to Tomura, Dabi mocked him by calling him a hollow cosplayer, implying that the latter enjoys insulting Spinner for laughs. 
Like the rest of his comrades, Dabi doesn't really care about Compress that much. Despite this, Dabi and Compress seem to have good camaraderie as they've teamed up on a few occasions. One of the notable teamups involved them both using their Quirks to take down Snatch when confronting Overhaul. 
However, Dabi enjoys insulting Compress when the opportunity comes by, notably when he calls the latter out for doing nothing during the clash against the Meta Liberation Army. 
Dabi has shown to be distrustful towards Hawks. It is unknown if he would have killed Hawks in his weakened state, despite that the two had set up High-End's attack together. Dabi even questions Hawks' integrity to the League of Villains as Hawks explained he had to keep up appearances for the Hero society.
Dabi lets him join the Paranormal Liberation Front, because he saw Hawks being able to commit a murder. It is unknown if Dabi sees through Hawks’ lies, but it's hinted that he's still observing him. Their relationship is mocking, dishonest and resembles a constant power struggle. As far as is known, Dabi is the only villain who may suspect Hawks of being a double-agent.
Dabi confronted Endeavor after the hero defeated High-End. Dabi greeted him and was amused by the latter's unfamiliarity with him while he seemed to know a lot about him.
Dabi earned the heroes' disgust after he callously disregarded the murder of another hero. Dabi tried to attack the weakened Hero, but retreated after Mirko appeared on the scene, though he told Endeavor that they would meet again while angrily calling him by his true name.
Dabi is deeply inspired by Stain's ideology. He joined the League as way to make Stain's ideals a reality. It is likely Dabi's decision to use an alias instead of his real name was inspired by Stain.