Short answer: no. You don’t need to have played Tales from the Borderlands before playing Borderlands 3. But it helps.
WARNING: A few minor spoilers are ahead if you haven't played Tales from the Borderlands. But at this point, they really shouldn't be huge surprises.
Chronologically speaking, Tales from the Borderlands takes place after Borderlands 2, but before Borderlands 3. This means there are events in Tales from the Borderlands that will come up in Borderlands 3, so you might be a bit confused if you’ve gone straight from Borderlands 2 to Borderlands 3. However, they’re not too hard to piece together on your own simply by playing Borderlands 3. Alternatively, you can just get a quick synopsis on Tales from the Borderlands online somewhere.
Tales will do a better job of introducing you to two characters that return in Borderlands 3, however. Rhys is a former Hyperion middle-manager that is the star of Tales from the Borderlands. His story mostly revolves around how he’s trying to track down another vault in order to regain his standing in Hyperion. Along the way, he meets the digital ghost of Handsome Jack, the villain of Borderlands 2, and also crashes the Helios space station into Pandora.
Vaughan is another recurring character that shows up first in Tales from the Borderlands. Rhys’ best friend, Vaughan and Rhys eventually part ways after the fall of Helios, where the former accountant goes off to become the head of his own bandit clan on Pandora. He’s more of a straight-man in Tales from the Borderlands, but that’s mostly because Tales is a bit more serious than the mainline Borderlands games.
While it’s nice to get both of their backstories in Tales, it’s not strictly speaking necessary. First, the broad strokes of Tales can also be found in the free Borderlands 2 DLC, Commander Lilith and the Fight for Sanctuary. In that DLC (which you got free so long as you already owned Borderlands 2 on Steam), you find out that Helios has crashed into the Pandora, Scooter is dead, the original Sanctuary has exploded, Vaughan is a weirdo bandit lord, and everyone is very excited to travel through space and see other planets.
Also, Rhys is voiced by Troy Baker in Tales from the Borderlands, but due to various Gearbox-related chicanery, he doesn’t reprise his role in Borderlands 3. That can be a little jarring for some players considering how iconic Mr. Baker’s voice has become in his various roles in various games.
On top of everything else, picking up Tales from the Borderlands is a little tricky right now. You see, Telltale Games famously closed its doors last September and pulled all their games from digital storefronts. Only just recently has Telltale announced they’re coming back under totally new ownership and plan to re-release all their old titles, but that might still take a little while. So even if you wanted to play Tales, right now you can't unless you already own the game.
If you absolutely must see Tales from the Borderlands before you play Borderlands 3, YouTube Let’s Plays are your best bet.