This may be more hassle than you bargained for but when I've run across similar problems, I'd measure the battery current (in milliAmps) under different operating conditions. For example, measure current when: supposedly turned OFF, on doing "nothing", back-light on/off, etc. Then compare to a different remote that doesn't pre-maturely drain the batteries.
The trick is you need to break a connection in the battery circuit to insert the meter leads. In the photo, I show a thin piece of 2-sided copper circuit board. There is a thin insulating layer so the two outer conducting layers are electrically isolated. Wires are attached to the two outer layers. This piece is inserted in the battery pack. This allows you to measure the DC current using any meter so-equipped. Obviously if you simply "short" the two wires then it's normal operation.
Most guys don't have scrap thin two layer circuit board material lying around...but I've made similar test "fixtures" using adhesive copper-foil on a thin piece of cardstock (insulating layer)...or even aluminum foil if you can make the electrical connection to the meter probes.
In any case, I'd be really curious about the "OFF" current as was suggested by someone earlier. When shut off, a circuit like this should draw less than 1 uA (microAmp).
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Separately, if your only measuring tool is a voltmeter, I suppose one could concoct some scheme to get a crude quantitative measure of what's going on. For example, you could measure the same battery in the suspect remote and a known-good remote under the same conditions. The idea is the measured battery voltage will be lower if the current draw is higher...due to the internal resistance of the battery. Again, it's crude and arguably bordering on a snipe hunt (a.k.a. wild-goose chase), but like they say - if your only tool is a hammer, all problems look like nails!
