No. Logaholic only needs the log file to import any new data once. After that, all the information is stored in the database, so the original log file data is no longer needed. Only NEW log file data is read from a log file after that.
In other words, you can safely remove or archive old log files that Logaholic has already analyzed.
Of course, Logaholic will still need to read a log file containing any new data each time it updates, so this is only relevant if you have a system that rotates your logs into separate files periodically.
You can even use Logaholic to recreate the log file later (Check the Profile delete options, Export to Log file format), in case you want to re-import re-install or use the information in another software package. But, any elements you filtered when importing in logaholic (like images) will be lost.
For this reason, it is always a good idea to keep a backup of the original log file data, you never know when you might need it again.