Do LI batteries catch fire? Well first you need to understand that there are multiple types of LI technology, Some are safer than others and I'm not expert enough to say which is which..
2nd: Over the years I've owned many computers (laptops and desktops) one which I bought new as a very nice IBM Thinkpad before IBM sold the name to Leveno
IT was a "Cool New Laptop" only that specific model came with a Made in China LI battery (Most all LI batteries sold in the US are made in China, for very good reasons, only two countries have supplies of LIthium and the other one is unfriendly to the USA) and it turns out I got a free replacement battery because some folks who bought a Cool New Laptop. Found out they had a very HOT new laptop. So hot it was on fire.. Due to the battery.
SO yes. some of them do catch fire.
I just don't know which ones