What is the best practice to backup a lucene index without taking the index offline (hot backup)?
This answer depends upon (a) how big your index is and (b) what OS you are using. It is suitable for large indexes hosted on Unix operating systems, and is based upon the Solr 1.3 replication strategy.
Once a file has been created, Lucene will not change it, it will only delete it. Therefore, you can use a hard link strategy to make a backup. The approach would be:
The cp -lr will only copy the directory structure and not the files, so even a 100Gb index should copy in less than a second.