Recently I came across following error while processing cube and after some research I was able to resolve this error. I thought its better to share the solution if someone come across same issue.
OLE DB error: OLE DB or ODBC error: The SELECT permission was denied on the object 'TableName', database 'DBName', schema 'dbo'.; 42000.
If you come across such error then open SSMS instance which cube is using as a underlying relational database server. Go to "Security" folder and then Logins. If your SSAS services are running under "NT AUTHORITY\NETWORK SERVICE" user then double click on that user under Logins.
Under "Login Properties" window, click on "Server Roles" and select role "sysadmin". Click ok and you have resolved your issue. check again with processing cube.
OLE DB error: OLE DB or ODBC error: The SELECT permission was denied on the object 'TableName', database 'DBName', schema 'dbo'.; 42000.
If you come across such error then open SSMS instance which cube is using as a underlying relational database server. Go to "Security" folder and then Logins. If your SSAS services are running under "NT AUTHORITY\NETWORK SERVICE" user then double click on that user under Logins.
Under "Login Properties" window, click on "Server Roles" and select role "sysadmin". Click ok and you have resolved your issue. check again with processing cube.