During the 2018 Arrowverse crossover, "Elseworlds," Oliver Queen (Stephen Amell) didn’t believe Batman was real out of professional jealousy, wanting to believe that he was the first big-time vigilante as the Green Arrow. After years of the Arrowverse not using Batman's lore, except for some of the Dark Knight's supporting characters, during the "Elseworlds" crossover in 2018 they finally acknowledged that Batman was a part of their world’s history.
Throughout its eight-season run, Arrow borrowed heavily from the Batman mythos. Oliver Queen acted a lot like Batman if he killed his enemies. He had the tragic backstory and focus on street-level crime, while also fighting many of Batman's villains, including lifting the Ra’s al Ghul storyline from Batman comics. This set up Oliver Queen as the Arrowverse equivalent to Batman (though Green Arrow has been “Batman with bow and arrow” since the 1940s), making things awkward when Batman actually is part of the story.
Oliver's denial of Batman comes from a personal place, believing that becoming a vigilante at the beginning of Arrow made him the first. As they’re heading to Gotham City to investigate the body swap that started the Elseworlds crossover, Oliver lays out his thoughts on Batman: “He’s not real. He’s an urban legend concocted by the Gotham Police Department to scare criminals. I’m the original vigilante, Barry.” Barry Allen (Grant Gustin) and Kara Danvers (Melissa Benoist) can’t believe what they’re hearing, alternating between amusement and shock at the fact that Oliver has seen so many things, but doesn’t believe in another street-level vigilante.
To be charitable to Oliver, Batman was never his focus. It’s unclear when Batman started his career, but Batman was known in 2003 when he failed to save Beth and Gabi Kane, as shown on Batwoman. This was years before Oliver's shipwreck started his path to becoming Green Arrow, but how much did he care about a vigilante in a city many miles away at that time? Then he’s on an island, in Hong Kong, or in Russia for five years, only coming back to deal specifically with the problems of Starling City. It’s mentioned that Batman disappeared three years before "Elseworlds," which would place his disappearance in late season 3 or early season 4 of Arrow when Oliver’s mission got bigger than just one city. There was never a time that it’d make much sense for him to focus on Batman.
Of course, Oliver makes his egotistical reasons for not believing in Batman obvious. Oliver wants to be the first vigilante, and Batman takes that away from him, starting his mission about a decade before Oliver starts his. Batman and Oliver are street-level vigilantes taking on corrupt people in a city that’s falling apart. Instead of being the first real vigilante, this makes Oliver look like a copycat, not someone with a unique, personal mission.
Oliver doesn’t believe there’s a Batman out of pride, wanting to be the first costumed vigilante rather than just a copycat of another hero in Gotham. It’s petty instead of a rational assessment of the situation since Oliver doesn’t want to believe that there’s anyone else like him out there even though Batman and Green Arrow are incredibly similar in the Arrowverse.