If the Raspberry Pi B+ is not receiving enough power and I try to use it, will it damage my Pi or peripherals?
I doubt it.
If it is writing to the SD card then the card filesystem may be damaged.
The most likely action is a Pi reboot.
We don't know what peripherals you may attach. You would have to look at them on a case by case basis. I reckon there would be great big warning labels on such a peripheral if it was susceptible to such damage.
(Edit) (It may be possible but probably not likely. What happened to me was that I had a bad HDMI cable)
I think it may be possible to damage the board its self. but it would have to be investigated further.
I connected my Pi to a 5v USB battery pack, Immediately the Pi began to show errors like no sound and video distortion on static pictures, Flickering and monitor turning off and on.
I then swapped the power supply to the one I've been using without problems and it did it again exactly like it did with the battery pack.
I then swapped the SD card to a known good one with the OS already installed and it did it again.
The only way for that to happen to my knowledge is because the Pi itself is messed up and is corrupting the cards.
I then put one of the messed up cards into another pi and it did the same thing. I'm hoping that it didn't permanently mess up the other pie or the SD cards.
I'm going to go get another pi and I want to investigate further with the same SD cards but I'm afraid to mess up another pi. So I think I'm just going to start over.