SELECT ... ORDER BY <boolean condition>

Having this situation:

SELECT DISTINCT "FieldName" 
FROM "TableName" 
ORDER BY "FieldName" ASC;

I'd like to have the lines containing '%|%' first followed by the ones without '%|%'.
How to achieve this?

Use a boolean expression on the first position of your order by list. Since true is 1, it's larger than a 0/false so you need to make that descending:
_{demo at db<>fiddle}

select*from(select distinct "FieldName" 
            from "TableName") as subquery
order by "FieldName" like '%|%' desc
       , "FieldName" asc;

A select distinct requires its order by to use the exact fields that are being selected, so it needs to be nested in a subquery before ordering under a regular select outside.
To avoid it, you can switch to a distinct on (as suggested by @Erwin Brandstetter), or a plain group by - after all, it'll extract distinct groups:

select "FieldName" 
from "TableName"
group by "FieldName"
order by strpos("FieldName",'|')>0 desc
       , "FieldName";

If you're just looking for a | character, a much more limited but simpler and faster strpos() should outperform pattern matching with like '%|%', similar to '%\|%' and ~ '\|'.