PL/SQL exception not reached, function doesn't return

Untested, but I would probably use a local variable to check whether my cursor loop found any rows, and check that at the end. (It could be a Boolean or a counter or something else. I've used a counter below.)

create or replace function is_artikl_eligible_for_zamjena
    ( insifraartikla in a_zamjena_artikli.sifra_artikla%type
    , indatumpocetak in varchar2
    , inskladiste    in a_zamjena.skl%type
    , inprodavaonice in a_zamjena.popis_prod%type )
    return number
is
    rows_found pls_integer := 0;
begin
    dbms_output.enable();

    -- check if there is sifra_artikla in a_zamjena_artikli where a_zamjena.datumKraj != null or a_zamjena.datumKraj > sysdate
    for i in (
        select azam.broj_zamjene
             , azam.datum_kraj
             , coalesce(azam.skl,'0') as skl
             , coalesce(azam.popis_prod,0) as popis_prod
             , azamar.sifra_artikla
        from   a_zamjena azam
               join   a_zamjena_artikli azamar
               on     azamar.broj_zamjene = azam.broj_zamjene
        where  azamar.sifra_artikla = insifraartikla
    )    
    loop
        rows_found := rows_found +1;

        if i.datum_kraj < sysdate - 1 then
            --level 0
            dbms_output.put_line('Datum kraj postojeće zamjene nije null ali je završio');
            return 1;

        else
            if i.datum_kraj is null or i.datum_kraj > sysdate then
                --level 1
                dbms_output.put_line('Zamjene imaju preklapajuće datume, provjeravam da li su na istim skladištima ili prodavaonicama');

                if i.skl != inskladiste or i.popis_prod != inprodavaonice then
                    -- level 2
                    dbms_output.put_line('Zamjene imaju preklapajuće datume ali nisu na istim skladištima ili prodavaonicama');
                    return 1;

                else
                    -- level 2
                    return 0;
                end if; --level 2
            else
                --else za datum kraj, level 1
                dbms_output.put_line('Zamjene nemaju preklapajuće datume');
                return 1;

            end if; --level 1
        end if; --level 0
    end loop;

    if rows_found = 0 then
        raise no_data_found;
    end;
exception
    when no_data_found then
        dbms_output.put_line('nema retka');

        return 1;

end;

I'm not sure raising a no_data_found exception is a particularly good approach here, though. In fact I'm not keen on a return inside a loop either. Maybe the problem goes away if you just assign the value you want to return to a local variable during the loop, and check it at the end, handling the case where it has no value.

With a local variable and a single return point instead of manipulating exception handlers, it would look something like this:

create or replace function is_artikl_eligible_for_zamjena
    ( insifraartikla in a_zamjena_artikli.sifra_artikla%type
    , indatumpocetak in varchar2
    , inskladiste    in a_zamjena.skl%type
    , inprodavaonice in a_zamjena.popis_prod%type )
    return number
is
    l_result number;
begin
    dbms_output.enable();

    -- check if there is sifra_artikla in a_zamjena_artikli where a_zamjena.datumKraj != null or a_zamjena.datumKraj > sysdate
    for i in (
        select azam.broj_zamjene
             , azam.datum_kraj
             , coalesce(azam.skl,'0') as skl
             , coalesce(azam.popis_prod,0) as popis_prod
             , azamar.sifra_artikla
        from   a_zamjena azam
               join   a_zamjena_artikli azamar
               on     azam.broj_zamjene = azamar.broj_zamjene
        where  azamar.sifra_artikla = insifraartikla
    )    
    loop
        if i.datum_kraj < sysdate - 1 then
            --level 0
            dbms_output.put_line('Datum kraj postojeće zamjene nije null ali je završio');
            l_result := 1;

        else
            if i.datum_kraj is null or i.datum_kraj > sysdate then
                --level 1
                dbms_output.put_line('Zamjene imaju preklapajuće datume, provjeravam da li su na istim skladištima ili prodavaonicama');

                if i.skl != inskladiste or i.popis_prod != inprodavaonice then
                    -- level 2
                    dbms_output.put_line('Zamjene imaju preklapajuće datume ali nisu na istim skladištima ili prodavaonicama');
                    l_result := 1;

                else
                    -- level 2
                    l_result := 0;
                end if; --level 2
            else
                --else za datum kraj, level 1
                dbms_output.put_line('Zamjene nemaju preklapajuće datume');
                l_result := 1;

            end if; --level 1
        end if; --level 0
    end loop;

    if l_result is then
        dbms_output.put_line('nema retka');
        l_result := 1;
    end;

    return l_result;
end;

I'm not sure what the dbms_output.put_line calls are doing, but that's another question.

You simply cannot raise a NO_DATA_FOUND exception using a for loop. You can do using a goto statement for insatance. See below example.

DECLARE
   v_attr char(88);
CURSOR  SELECT_USERS IS
 SELECT id 
 FROM USER_TABLE
 WHERE USERTYPE = 'X';
BEGIN
    FOR user_rec IN SELECT_USERS 
     LOOP    
        BEGIN
            SELECT attr 
            INTO v_attr 
            FROM ATTRIBUTE_TABLE
            WHERE user_id = user_rec.id;            
         EXCEPTION
            WHEN NO_DATA_FOUND THEN
               -- user does not have attribute, continue loop to next record.
               goto end_loop; ---Adding a label
         END;

        <<end_loop>> --Label
        null;         
    END LOOP;
END;