Tears of the Kingdom's new trailer shows what looks like a makeshift car, kind of like the awkward prop mess you'd put together in Garry's Mod. But that raises a question - what happened to Link's motorcycle? It was DLC granted, but that's sparked fans to try and come to a rational explanation as to why it's suddenly disappeared, leaving Link to play with scrap.
One of the more logical explanations is that you summoned the bike via the Sheikah Slate, which Link no longer has. In the trailer, we see him using his new arm to lift objects, seemingly replacing its function, while there's no sign of the Slate on his person. Ergo, no bike.
We've solved the mystery, right? Maybe, but there's more to it. As a few commenters pointed out, at the end of Breath of the Wild, all tech seemingly deactivated, such as Vah Ruta. The Sheikah Slate may have shut down with the Divine Beasts, leaving Link up a creek (or upfield) without a paddle (or in this case, a peddle).
What this means is that Link may not have willingly given up his Sheikah Slate, but rather he was forced to given that the tech stopped working. However, Zelda appears to have her own in Tears of the Kingdom, which we saw at the end of Breath of the Wild, so it's unclear how solid that theory is. Maybe they found a new way to get them working, or built one from scratch.
Moving away from in-universe reasons and lore, several suggest that it's simply because the bike was DLC and that Nintendo wants to, like with any sequel, start the player over fresh. That means no gear or upgrades, so no bike. It's a little boring, but that's the out-of-universe answer.
There are some fun theories, though. One suggests that we'll simply get a pop-up message saying "Your Master Cycle Zero Broke" and that'll be that. Maybe Beedle took it for a joy ride and crashed it in a ditch. Another theory, from PixelatedFrogDotGif, posits that Tears of the Kingdom will open up with a 20-minute montage showing the bike crash in a "radical explosion".
SpaceCube00 says that "Ganon was jealous of Link's sick bike so he took it" - that certainly tracks. But perhaps my favourite of all the theories is that there are new Hyrule traffic laws that limit the driving age for using motorcycles - if you're too old, you can't ride. Given that Link is well over 100 at this point, he fits the bill, and so his license is null and void. The plot would then revolve around Link trying to change the traffic laws by overthrowing the kingdom. That might be a bit far-fetched, though.
All of these are great answers, but I'm going to just assume he took it with him to one of the Mario Kart tournaments and lost it. Bowser might have it in his castle somewhere on display. That sounds right, doesn't it?