Hello. I know this is an old thread but I wanted to respond in case anyone has the same problem and finds this thread.
I had the exact same issue with the power button. Luckily, the DIY fix is pretty simple. In my case, the component that failed was a tiny tactile switch--basically just a concave disk that snaps up and down when pressed, making contact with the pad below it. Corrosion (from water I guess) had build up underneath the disk, making it impossible to press down and make contact with the pad. The solution was to take it apart, clean the corrosion out, and reassemble, which fixed the problem.
In detail:
1. Take apart the DR-07 until you have free access to the offending switch. It will look like a small square component (5x5mm ish) with a circle inside it, covered with an orange adhesive film. The DR-07 disassembles by removing a bunch of screws, both on the outside and inside. The only trick is that the battery contacts (the ones near the middle of the unit, not the ones at the bottom) are soldered to the circuit board. To remove the back of the case (where the batteries sit) you have to remove the screws, then pull up on the back while pushing the battery contacts down so they slide out of the case. It'll make more sense once you look at it. It's kind of finicky, just be patient, don't yank the case apart or you will pull them off and then you'll have to pull out the solder. Note that the "hold" switch is not held in by anything, it's just sitting in there, so it will fall out when you separate the case. Just note which way it goes in.
2. Once you've determined which switch is causing the problem (it won't "click" when you press on it like all the other similar ones do), simply remove the orange adhesive film with a pair of tweezers, then remove the metal disc (which is free-floating). Clean the disc and the surface below it. I used an old toothbrush, a toothpick, and a 50/50 water/vinegar solution. After cleaning, dry it very thoroughly. I recommend a blow dryer or letting it sit an hour or so. (If you use a blow dryer make sure you don't lose the disc when it flies away!)
2.5. I had a bunch of other corrosion on my circuit board so I cleaned and dried it at this stage as well. Not sure why it was so bad.
3. Replace the disc and adhesive film. If you ruined the adhesive film taking it off just make a little square out of scotch tape and use that.
4. Reassemble in the reverse order.
Hope this helps. Let me know if you need more specifics.