implode can only implode an array into a string, it can't "add" to a string or array.
In your case you want to loop through each record, and append it to an array,
and then implode that array.
$sql ="SELECT ID FROM likvid WHERE DATUM='$date' and NAMN='$name'";
$result = mysql_query($sql,$db);
$aIDs=array();
while ($row=mysql_fetch_array($result)) {
$aIDs[]=$row['ID'];
}
$sIDs = implode(',',$aIDs);
$sql="SELECT * FROM table WHERE ID in [$sIDs]";