After moving a DB to a new disk, the log file isn't dumping data to the DB
However, two days later, the .mdf file is the same size as
  pre-transfer (I took it offline and only the timestamp changed).


There was enough free space in your data file for any new data that was written. When that spare/free space is consumed the file will grow, not before. You can confirm the size and free space in your data file with:

SELECT 
    [File Name] = name
    , [Physical Name] = physical_name
    , [Total Size in MB] = size/128.0
    , [Available Space In MB] = size/128.0
        - CAST(FILEPROPERTY(name, 'SpaceUsed') AS INT)/128.0 
FROM 
    sys.database_files;



  I checked the log, which is usually under 1GB, and found that it has
  now reached 30GB.


Your database is in full recovery mode and you aren't backing up the log. For this and all manner of transaction log growth explanations, see @MikeWalsh's canonical answer on the topic at Why Does the Transaction Log Keep Growing or Run Out of Space?