Q:

Primary key not equal zero reduce performance

Does primary key not equal zero id != 0 in where clause reduce query performance? Does this checking prevent taking the advantage of indexing?

A:

Does primary key not equal zero id != 0 in where clause reduce query
  performance?

It does in the way that you end up with a full table scan.

Does this checking prevent taking the advantage of indexing?

Yes. Since id != 0 will almost always evaluate to true you end up retrieving all of the table data. 
The less % of the results you're returning, the more likely it is for Oracle to take advantage of your index. Your condition is returning all of your data therefore it isn't taking advantage of your index.
If you're wondering whether you can improve your query performance with the condition != 0 the answer is you can't. It's already good enough. Sometimes you have to accept the full table scans, you can't always take advantage of your index.

