Well in a sense *all* birds are dinosaurs, just highly modified ones...
But no, the dodo wasn't more closely related to dinosaurs than any other bird, and it *was* a bird.
It went extinct in the mid 1600's, but that's still very modern times compared to the age of the dinosaurs.
The question, of course, is how do we know it was a true bird? The answer is pretty interesting.
You'd think that we'd have all sorts of dodo artifacts gathering dust in museums around the world, but in fact is that since the dodo went extinct so long ago, we actually have very few remains of it left. In all the world, that we know of, there exists just one preserved dodo head, one preserved foot, one complete skeleton, and a hundred or so miscellaneous bones in various condition.
Note that that's a classic "bird" breastbone on that skeleton, it's not dinosaurian. And so on.
However, using bits from these items, researchers (Dr Alan Cooper and Dr Beth Shapiro from Oxford's Henry Wellcome Ancient Biomolecules Centre, Dr Dean Sibthorpe, Andrew Rambaut, Dr Graham Wragg, Dr Olaf Bininda-Emonds and Dr Patricia Lee from the Department of Zoology, and Dr Jeremy Austin from the Natural History Museum, London) extracted and amplified dodo DNA, and compared it with the DNA from the Rodrigues Island Solitaire, a nearby (also extinct) dodo-like bird, as well as 35 species of pigeons, doves, and other bird species. Why pigeons? Because earlier analysis of the dodo skeleton showed that it was most similar to that of the pigeon family. This isn't as odd as it might sound if you're only familiar with small American pigeons...
The DNA analysis showed that as expected, the dodo was most closely related to the Solitaire, but beyond that its nearest relative was the Nicobar pigeon from southeast Asia:
Next nearest was the crowned pigeon of New Guinea:
And the tooth-billed pigeon of Samoa:
The similarity to the dodo's beak should be obvious.
Interestingly, all three birds are ground-feeders, with strong hooked bills for cracking hard nuts. The dodo's extreme bill was used for feeding on the nuts of the Calvaria tree, which were very hard.
The researchers wrote:
'The genetic differences suggest that the ancestor of the Dodo and Solitaire separated from the Southeast Asian relatives around 40 million years ago, and sometime after this point flew across the Indian Ocean to the Mascarene Islands. The data suggest that the Dodo and Solitaire speciated from each other around 26 million years ago, about the same time that geologists think the first (now submerged) Mascarene Islands emerged. However, Mauritius and Rodrigues islands are much younger (8 and 1.5 million years respectively), implying that the Dodo and Solitaire used the now sunken island chain as stepping-stones. Furthermore, the isolation of Rodrigues Island suggests that the Solitaire, at least, may have still been able to fly as recently as 1.5 million years ago.'
There is actually a project underway to try to "bring back" the dodo in a manner very similar to "Jurassic Park", by extracting as much DNA as possible from remaining dodo parts and then merging them with pigeon DNA to fill the gaps and then cloning. It's a longshot, of course, but it's an interesting attempt, and it's got a better chance than bringing back dinosaurs, since 1600's DNA is a lot "fresher" than 65+ million-year-old dinosaur DNA.
Another interesting fact about the dodo is that although most people think of them as very fat, they may have been much slimmer in the wild. They were large birds, about the size of a large tom turkey, but the "fat" drawings of them were all from specimens brought to Europe, while drawings of dodos in their native Mauritius were much slimmer. The thought is that the captive dodos had been overfed during their captivity, leading to the "fat look" in many drawings:
(Having gone extinct in the 1600's, of course, there are no photographs of living dodos.)
But bone structure calculations, reconstructions from skeletons, and sketches from indigenous dodos has led to a slimmer view of the dodo:
They still might have gotten plumper at certain times of the year, though. Sketches from different seasons seem to show the dodo at different weights, and it has been hypothesized that since the dodo's nut foods were seasonal, they may have packed on weight in times of plenty to get them through the scarcer months of the year. This may also explain why they fattened up so easily when overfed in captivity.
It further might explain why they lost the power of flight. Given a) their island had no natural predators, b) the plumper, heavier birds may have survived the winter better, and c) a bigger bird has a bigger stronger beak for cracking hard nuts, it may well be that the dodos had more evolutionary incentive to get large and therefor flightless than they did to retain the not-so-useful-to-them ability to fly.