The only way that you can damage your speedlight from rechargeable batteries is by leaving them inside there and them leaking acid... thats it. Well, that and using your flashes in an unreasonable manner!
What is more important, is how you were using your flash to use up 2 sets of batteries in a short time. I think that your expectations of how many times you can use them within a period and more importantly, at what power... needs attention.
Yes, batteries will get hot if used, no you cannot blow your flash... unless you
really over do things.
First, I can get an easy 400-500 flashes out of a set of 2650mah energizers... but then again, I never try to go above 1/2 power. Raise your ISO! Use multiple flashes! Don't expect your speedlights to be set to full power and run off a couple of sets of batteries to last you very long.
If you had set your speedlight to full power and popping off one shot after another at 4 second intervals, wow, thats really abusing your flashes. Most are not made for that. A thinking photographer will set the flash to 1/4 power and pump up the ISO from 100 to 400 and get IDENTICAL exposures.
At 1/4 power, you've basically quadrupled the number of flashes you can take, and you've dropped recycle times from 3-5 seconds to 1-2 seconds, and could go all night like that.
Lithium batteries will NOT recharge your flash any faster, not give you more flashes per charge, are not rechargeable and are bad for the environment. Why are you not using rechargeable batteries in the first place?
At an average wedding, I can pop off 1200-1500 pictures, 90% of those are with flash, and I have YET to get through my 2nd set of batteries. That alone should tell you something. Either you are using some terribly crappy batteries and/or your methodology for using the flah(es) needs changing.