Soo. I might have given Lucifer a redesign 

I currently have a thing for furry elf ears. 

Anyways- I like this design more, I didn't change much besides colors and horns, plus it fits into them originally looking like heaven-born angels before Lucifer fell.

SO, LuLu, his tears burn him : ) 

It is canon that Lucifer has the hottest fire in Wonderland, tying with Mysterio, it is hotter than the sun and his body heat already feels like touching hot metal without him purposefully cooling himself down. And now for me trying to do a stupid explanation. 

When Lucifer starts to get upset and cry his body temperature increases dramatically, it takes a few moments for tears to actually even start to form since every time a tear forms and evaporates it cools that area off however it also burns him, so each tear that gets further down burns him and cools down areas for more tears to survive and the process cycles, while that process is cycling he also starts to bleed from the wounds and makes it look like he is crying blood it also makes a very satisfying sizzling sound and smoke starts to form

To answer a question that might be asked; Does water normally hurt him?

I wanna say, no, he has the ability to control his body temperature, he always had and has been forced to take baths and drink water, yes it is possible to use water to against him however you have to be careful and wait till he has low body temperature, as he can easily melt ice or evaporate water. When fighting Lucifer it's generally best to use water or ice to distract him.