Linked Server along Oracle 32 and 64 bit Oracle Client
So when I run the job and it is connecting Oracle Server, will it use Linked Server or will directly use Oracle 32-bit client?


Depends on how the DTS package is written, since DTS was the old 2000 era packages I have no idea if they imported a connection in the package or query through the linked servers. You'd have to investigate to see if it references the linked server using TSQL or it uses a connection, directly.


  If linked server is failing will that can be the reason for job failure. Error message when Testing Linked Server is: ORA-12154: TNS: could not resolve the connect identifier specified


Looks like you should check the tnsnames.ora file to make sure the listeners are identified properly.


  Is the order in which oracle bin path is defined in PATH System Environment variable can impact this issue in anyway?


Potentially, depends on how/what the package is doing.


  After making the changes in PATH System Environment variable do we need to restart SQL Server service or entire server.


If you make changes to the system variables it's best to restart the server.