The likely cause for this is that you might have more than one index named xyz on your database. Your IF EXISTS SELECT statement is not taking into consideration which table xyz is on.

You can check for this condition yourself by running the select statement by itself.

Try changing your query to the following to limit the scope:

If Exists 
(
    Select * 
    From   sys.indexes 
    Where  name = 'xyz' 
    And    Object_Id = Object_Id('dbo.Table_Name')
)
Drop Index xyz On dbo.Table_Name;

Is there any way to tell SQL to stop using an index?

That would be disabling an index:

alter index IX_YourIndex
on dbo.YourTable
disable;
go

But that, too, will be running into concurrency issues on a highly transactional object, as that operation above still requires a schema-modify (Sch-M) lock on the object.

Or is it possible to run a query that will attempt to obtain an exclusive lock on the index, wait indefinitely, and then delete the index?

That would be the default behavior of SQL Server. The only "command timeout" is going to be enforced by client-side software. I'm going to assume that you're using some sort of GUI to try to delete/disable the index (SSMS maybe?). Either way, if you were to put this in a new query window (SSMS), it would wait indefinitely for the locked resource to become available so it can be granted the necessary locking:

drop index IX_YourIndex
on dbo.YourTable;
go

This is one of the reasons why it's almost always easiest to do these types of operations during a slow time, or a maintenance window. That typically can ensure little to no user activity, making these simple tasks...simple and quick.

And if you want to see what/who your drop index request is waiting on, you can run a query similar to this:

select
    session_id,
    command,
    wait_type,
    wait_time,
    blocking_session_id,
    wait_resource
from sys.dm_exec_requests
where command = 'drop index';