This is a complicated one, and I think it is hard to provide a 100% sure answer without actual table and data to play with.
However let me try to describe the logic, and build the query step by step so you can paly around with it and possibly fix lurking erros.
1) We start with selecting all ip adresses that posted more than 60 votes on a given day. For this we use a group by on the voting day and on the ip adress, combined with a having clause
select date(voted), ip_adress
from table 
group by date(voted), ip_adress 
having count(*) > 60
2) From then, we go back to the table and select the first 60 ids corresponding to each voting day / ip adress couple. id is an autoincremented field so we just sort using this field and the use the mysql limit instruction
    select id, ip_adress, date(voted) as day_voted
    from table 
    where ip_adress, date(voted) in (
        select date(voted), ip_adress 
        from table 
        group by date(voted), ip_adress 
        having count(*) > 60
    ) 
    order by id
    limit 60
3) Finally, we go back once again to the table and search for the all ids whose ip adress and day of vote belong to the above list, but whose id is greater than the max id of the list. This is achieved with a join and requires a group by clause.
select t1.id 
from 
    table t1
    join (      
        select id, ip_adress, date(voted) as day_voted 
        from table 
        where ip_adress, date(voted) in (
            select date(voted), ip_adress 
            from table
            group by date(voted), ip_adress
            having count(*) > 60
        )
        order by id
        limit 60
    ) t2 
        on t1.ip_adress = t2.ip_adress 
        and date(t1.voted) = t2.day_voted and t1.id > max(t2.id)
group by t1.id
That should return the list of all ids that we need to delete. Test if before you go further.
4) The very last step is to delete those ids. There are limitations in mysql that make a delete with subquery condition quite uneasy to achieve. See the following SO question for more information on the technical background. You can either use a temporary table to store the selected ids, or try to outsmart mysql by wrapping the subquery and aliasing it. Let us try with the second option :
delete t.* from table t where id in ( select id from (
    select t1.id 
    from 
        table t1
        join (      
            select id, ip_adress, date(voted) as day_voted 
            from table 
            where ip_adress, date(voted) in (
                select date(voted), ip_adress
                from table 
                group by date(voted), ip_adress
                having count(*) > 60
            )
            order by id
            limit 60
        ) t2 
            on t1.ip_adress = t2.ip_adress
            and date(t1.voted) = t2.day_voted
            and t1.id > max(t2.id)
    group by t1.id
) x );
Hope this helps !