Does a read apply locks when run outside of a transaction? If implicit_transactions = off and I execute a SELECT query outside of an explicit transaction, will any locks be applied or do locks only make sense as part of a transaction?
The way SQL Server handles locks is based on the isolation level that you are using and the state of the transaction. The default isolation level for SQL Server is READ COMMITTED. So if you perform a SELECT command without an explicit transaction, then the dynamic locking will decide the lock granularity. 

For READ COMMITTED isolation level, the SHARED locks acquired on the index keys/RID (row identifier) will be released after the scan/seek has completed.  

Refer to Locking in SQL Server and
Isolation Levels