I really think Sonic has a chance here. Sonic is one of the fastest guys in the tourney - The only one I can think can beat him in a race would be Flash. But Sonic isn't just a fragile speedster, oh no. With his rings, he basically gets two chances at life. If he's hit once, he loses his rings. If Doom hits him again and he has no rings, only then would Sonic die. The thing is, Sonic could grab a ring before it disappears and regain his temporary immortality, which forces Doom to hit him again and again until Sonic is totally out of rings. So if you combine these two things, Doom would have to hit sonic at LEAST twice in order to take him out, and Sonic as previously stated is one of the fastest characters in the tourney. He could definitely dodge many of Doom's attacks.
On top of that, we have Sonic's two main forms of attack, the Spin Dash and the homing jump. The homing jump can home in on Doom from the air, striking Doom with sharp quills at a very high speed. This is something that has taken down massive machines, giant robots, and even cosmic entities as seen by Super Smash Bros. Brawl. As for the Spin Dash, it's much the same but a very fast ground attack. This could knock doom's legs from under him, allowing Sonic to strike hard and fast.
Obviously it'll take a lot of hits to take out Dr. Doom, but I think if anyone can do it it'll be Sonic the Hedgehog. It's not like he's a stranger to evil doctors.