First there's the issue of canon itself. In Trek, that's limited to what happens on screen. In Doctor Who, that's quite different and, honestly, the focus in Doctor Who is much more on telling entertaining stories than on maintaining continuity. That creates one issue, where one has a rigid continuity and the other doesn't.
At first glance, it seems possible the two could fit together, but on closer inspection, there are enough inconsistencies to show they cannot share the same canon.
I looked at timelines for both shows, one for Trek and one for Doctor Who. What surprised me was that it wasn't as easy to jump right in and find as many contradictory events as I wanted to find. Both seem to follow the same timeline in terms of when humans spread out to the stars.
I was going to also say there was no Eugenics War in Doctor Who, but that doesn't really hold water, since that was in the 1990s and that has been unofficially (as in "wink, wink, nudge, nudge") moved up since it conflicts with reality. That makes the Eugenics wars a not-useful plot point in proving incompatibility.
Some events may seem to dovetail, for instance, There's The Dalek Invasion of Earth (where Daleks invade Earth) in 2164, which would be right after the series Enterprise. You can't really prove they don't fit there, since we don't know for sure that there was no invasion of Earth after the series finished. But we do know that, by that time, the Federation was growing, so it's unlike Earth would be alone and needing to fight off the Daleks by itself if there were such an invasion.
There's also the issue that the world, as pictured in The Dalek Invasion of Earth is one without starships like the Enterprise and, considering that it would be so soon after the Xindi attacked Earth, the planet would certainly be able to do more from space to defend itself as seen in the story with the Doctor.
This, alone, should prove canon involving The Doctor violates Trek canon (or the other way around), since it provides two vastly different views of the same time period in Earth's future.
I did find that, in Doctor Who, there is The Waters of Mars, which is about the end of the first permanent colony on Mars. This takes place in 2059, during the "post atomic horror" in Trek. This was not a time, in Trek, when humans were reaching out to settle colonies on other planets and during the story in Doctor Who, there was enough communication with Earth to present the picture of a society that was not dealing with the aftermath of a terrible war. In fact, when the Doctor visits Earth to take one character back, Earth looks quite calm.
Also, there are no indications in either universe of the major threats in the other universe. For instance, Starfleet has never made reference of the Daleks or Cybermen, but they are major threats in many eras. The Doctor has never dealt with The Borg, in spite of having been around in the same approximate time.
So, while it's hard to prove, there are enough contradictory views of what was going on around Earth at the same time to prove the shows do not share canon. It's easiest to find this if one examines the 22nd century on both timelines.