ok, The voltage requirements of the diodes vary as has been stated before. But also you need to account for how much voltage the driver requires as well. there are boost drivers, and linear drivers. you can buy a rkcstr driver which is linear, or a flexdrive boost driver fairly easily, or make a driver yourself. Take a boost driver, hook it up to a LOC 16x red diode, and a linear driver and hook it up to a LOC 16x driver. set both diodes to 400mA. you end up with two diodes, at the SAME current level, but the flexdrive will run nice and happy off of 3 volts, where as the rkcstr driver will need around 5 volts. The flexdrive would power a blu-ray laser as well with only 3 volts. however the rkcstr driver would need 7.2 or more volts. again, same diodes same power, different voltage required by the different drivers. As a side note, you could feed the rkcstr driver 12 volts instead of 7.2 and it would be just fine. and 3 volts simply wouldn't be enough to make the diode start to lase.
Also, if you read through a couple diode testing threads such as this one
http://laserpointerforums.com/f50/flat-red-laser-diode-testing-2nd-phase-47044.html
you will see that a red diode needed 1.72 volts at 26mA, however the current was increased to 516ma and the voltage required went up to 3.14 volts. This is in the same diode. A LOC diode at the same power level needed 3.45 volts. so two different RED diodes, same current and two different required voltages...... You might get the idea it's not just a simple answer as the voltage required seems to vary considerably....
Also keep in mind a battery doesn't give an exact voltage either. I power my LOC with 2 3.7v lithium cells. (aaa sized) however fully charged they are putting out 4.2 volts. the voltage will considerably below 3.7 volts before they run out as well. (2.something)
Remember, the current in mA is more important then the voltage going to the diode. I doubt you will be running 20 or more volts through a diode anytime soon just powering it with regular batteries. So basically too few volts will not be enough to turn it on, and shouldn't hurt anything. where as you need more then a couple extra volts to hurt it. But the current in mA of course is a different story entirely. What killed your diode was way too much current. To decide how much voltage to power one, first select and purchase a driver, then look at how much voltage the driver will need...
As far as killing your red diode, what did you power it with? you did use a driver right? and what did you set the current to?
this thread has a lot of useful data. you can see a lot of info on specific diodes. Just keep in mind things like how much voltage is required DOES vary. and you don't need exact voltages.
http://laserpointerforums.com/f51/diodes-compilation-all-diodes-data-one-thread-45042.html