What is the azure table storage query equivalent of T-sql's LIKE command?

There's no direct equivalent, as there is no wildcard searching. All supported operations are listed here. You'll see eq, gt, ge, lt, le, etc. You could make use of these, perhaps, to look for specific ranges.

Depending on your partitioning scheme, you may be able to select a subset of entities based on specific partition key, and then scan through each entity, examining message to find the specific ones you need (basically a partial partition scan).

Another option would be export the logs from Azure Table storage to csv. Once you have the csv you can open this in excel or any other app and search for the text.

You can export table storage data using TableXplorer (http://clumsyleaf.com/products/tablexplorer). In this there is an option to export the filtered data to csv.