I have a test framework that I use to run different CPU assemblies. Just a power supply in a housing with a I2C connected 4x20 character LCD. On the input of the I2C LCD interface I have a level converter so I can safely connect the 5V LCD to a 3V module. (ESP8266)
When using the setup with a 5V micro-controller I have a jumper that applies +5V to the low voltage side of the level converter. I had this jumper still in place when I last used the setup with a ESP8266. I was doing a voltage measurement on a pin with a volt meter and I was getting 5V when I should have only read 3.3V. I thought the ESP module was the problem. After a bit of troubleshooting I located the short between +5V and +3.3V lines. And it was the jumper that I had forgot about. I pulled the jumper, applied power, and all was normal again.
But I had expected there would have been damage to the ESP module. I can't say there wasn't any damage but it appears there might not have been. But I still am surprised that it ran the code properly when I had it connected to +5V. It connected to my network. I was running the applications seemingly without error.
I have plans to use the ESP8266 with a battery power source. I wasn't sure what I was going to do about the higher than 3.3V battery power. Fully charged lithium batteries for example. Maybe I won't do anything. I don't recommend this, but I do think it is worth investigating.