I had a similar error when using classic ASP and Conn.Open "Provider=OraOLEDB.Oracle..." as is shown in the code above.
However, if I used Conn.Open "DSN=MyDSN;User ID=..." the connection worked.
In order to use the "Provider=OraOLEDB.Oracle..." syntax, I had to create a 32-bit application pool in IIS - create the application pool, then set Enable 32-Bit Applications = True (in advanced settings).
Note that 64-bit Windows has both 32 and 64-bit ODBC configurations. The default ODBC manager is 64-bit. Classic ASP applications can run in 64-bit mode and use 64-bit ODBC DSNs with some limitations like using Conn.Open "DSN=MyDSN;User ID=...".
If you use the Conn.Open "DSN=MyDSN..." syntax and you run in 32-bit mode, you will need to configure the DSN using the 32-bit ODBC manager. To access the 32-bit ODBC manager, run C:\Windows\SysWOW64\odbcad32.exe
The 32-bit and 64-bit ODBC manager windows look identical - there will just be different lists of DSNs that are configured.
One other tool that I found helpful is to create a file named DBConnection.udl. (The file name is not important - just the .udl suffix.) When you double-click the file, a Data Link Properties window will open and allow you to select the provider and connection info. There is also a test button to verify your connection. After setting the connection variables, close the window and open the DBConnection.udl file with a text editor (notepad.exe works) and it will display the OLEDB init string that can be used in your application.