First off, pairing a classic dob with a DSLR is a bit like a shotgun marriage. A dobsonian is fundamentally a visual telescope. Most manufacturers don't even consider the possibility that these instruments could be used for data collection via a sensor. There are 2 issues here:
1. The dobsonian is not tracking
The sky is moving, the dob stays still. You have to push the dob to keep up with the sky. Any long-exposure photo would be smeared. To remedy this, you'll need an equatorial platform, which will move the dob in sync with the sky.
Please note that only the best platforms allow reasonably long exposure times. Then the results can be fairly good.
2. There isn't enough back-focus
The best photos are taken when you remove the lens from the camera, plug it into the telescope directly, and allow the primary mirror to focus the image directly on the sensor. This is called prime focus photography. But most dobs can't reach the sensor within the camera, because their prime focus doesn't stick out far enough. There are several remedies for this, like using a barlow, moving the primary up, etc.
The bottom line is that it takes some effort to make a dob and a DSLR play nice together. Is it doable? Yes. Is it simple and immediate? No. So the literal answer to your question is that there isn't much you can do with just a dob and a DSLR.
You can take photos of the Moon and the Sun, because the short exposure there does not require tracking, but that's pretty much it. Here is an image of the Moon I took with a home-made 6" dob (with home-made optics) and a mirrorless camera (prime focus, about 1/320 sec exposure):
Makes a cute little desktop background, I guess, but it's definitely not research-grade.
Now add a tracking platform and things become more interesting, and the possibilities open up quite a lot.
In a more general sense:
There are telescopes that are specifically made for astrophotography. They have lots of back-focus, they are short and lightweight and therefore can easily be installed on tracking mounts. More importantly, there are tracking mounts made specifically for imaging - very precise, delicate mechanisms that follow the sky motion with great accuracy. In fact, the mount is more important than the scope.
A typical example would be a C8 telescope installed on a CGEM mount, or anything equivalent. Barring that, a dob with lots of back-focus sitting on a very smooth tracking platform (probably not as accurate as a GEM, but good enough for many purposes).
Make sure you don't exceed the load capacity of the mount. If the mount claims it can carry X amount of weight, it's best if the telescope weight doesn't exceed 1/2 of that amount. Close to the weight load limit, all mounts become imprecise.
Once you have: a tracking mount, a good camera, and a telescope (listed here from most important to least important), you can start imaging various portions of the sky for research. There are 2 main classes of objects that you could image:
1. Solar system objects
They're called "solar system objects" but the class includes anything that's pretty bright, not very big, and it's high resolution. Tracking is important but not that crucial.
You need a sensitive, high speed camera that can take thousands of images quickly (a movie, basically). These are called planetary cameras. As a cheap alternative in the beginning you could use a webcam, there are tutorials on the Internet about that. A DSLR in video mode in prime focus might work, but it's going to do a lot of pixel binning, so resolution would be greatly reduced unless you use a very powerful barlow (or a stack of barlows).
You'll load all those images in a software that will perform "stacking" to reduce them all to one single, much clearer image.
The scope needs to operate at a long focal length, f/20 being typical, so a barlow is usually required. The bigger the aperture, the better.
2. Deep space objects (DSO)
These are anything that's pretty faint and fuzzy, like galaxies, but some comets are also DSO-like in their appearance. You need to take extremely long exposures; usually a dozen or a few dozen images, each one between 30 sec and 20 min of exposure. Extremely precise tracking is paramount, so you need the best tracking mount you could buy. Autoguiding is also needed to correct tracking errors.
The scope needs to operate at short focal ratios, f/4 is pretty good, but as low as f/2 is also used; focal reducers (opposite of barlows) are used with some telescopes, like this or like this. Aperture doesn't mean much; small refractors are used with good results.
The camera needs to be very low noise; DSO cameras use active cooling that lowers their temperature 20 ... 40 C below ambient. DSLRs can also provide decent results, but their noise is typically higher than dedicated cameras, so you need to work harder for the same results.
Specific software is used for processing, stacking, noise reduction, etc.
So what can you do with such a setup?
Comet- or asteroid-hunting works pretty well. Terry Lovejoy has discovered several comets recently using equipment and techniques as described above. Here's Terry talking about his work.
Tracking variable stars is also open to amateurs. This could also be done visually, without any camera, just a dob, meticulous note-taking, and lots of patience.
With a bit of luck, you could also be the person who discovers a new supernova in a nearby galaxy. You don't need professional instruments, you just need to happen to point the scope in the right direction at the right time and be the first to report it. This also could be done purely visually, no camera, just a dob.
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