In general, how important is it to keep log files (*.log)? I seem to have tons of them for many, many different applications. Getting rid of them (or at least some of them) would free up some space on my relatively small hard drive.
Log files are extremely important - anything from your Operating System to your anti-virus contains some sort of *.log file. Not recommended. This is why apps like CCleaner brick systems.
I've yet to hear of any software (let alone an entire OS) whose operation depends on log files—most are error and install logs anyway, and typically when you delete them they will just be automatically created again and logging will continue if necessary. And since when has CCleaner ever bricked a system?
Only if you use them wrongly and/or are not careful and know what you are doing when you remove or delete stuff, basically like with everything else. Edit: and the default settings in CCleaner has not bricked any system that I have heard of.
CCleaner Leave checked: Options >Advanced >Only delete files in Windows Temp folders older than 24 hours.