Opening and Closing Balance Sum

You don't need the CTE in this case. It is redundant. You don't need to convert the date in string to date data type. Just specify the date in ISO format YYYY-MM-DD will do.

I am may be wrong but your expected data does not matches with the sample data.

Note : i excluded the ConfigurationCode in the query as i am not sure how that column play a part in your requirement.

DECLARE @Date_Fr    DATE    = '2018-08-25',
        @Date_To    DATE    = '2018-08-25'

SELECT  t.COATitle, 
        OpeningDebit    = ISNULL(o.OpeningDebit, 0), 
        OpeningCredit   = ISNULL(o.OpeningCredit, 0),
        t.DrAmount, t.CrAmount,
        ClosingDebit    = ISNULL(o.OpeningDebit, 0) + t.DrAmount,
        ClosingCredit   = ISNULL(o.OpeningCredit, 0) + t.CrAmount
FROM    (
            SELECT  COATitle, 
                    DrAmount    = SUM(CASE WHEN [TransactionDate] >= @Date_Fr THEN DebitAmount ELSE 0 END), 
                    CrAmount    = SUM(CASE WHEN [TransactionDate] >= @Date_Fr THEN CreditAmount ELSE 0 END)
            FROM    Transactions t
            WHERE   [TransactionDate]       <= @Date_To
            GROUP BY COATitle
        ) t
        OUTER APPLY
        (
            SELECT  OpeningDebit    = SUM(DebitAmount), OpeningCredit = SUM(CreditAmount)
            FROM    Transactions x
            WHERE   x.COATitle  = t.COATitle
            AND     x.[TransactionDate] < @Date_Fr
        ) o
WHERE   o.OpeningDebit  IS NOT NULL
OR      o.OpeningCredit iS NOT NULL

This may help. I put additional check, because I don't know if situation when opening balance is debit and closing is credit for example, is valid.

DECLARE 
    @from date = '25-08-2018',
    @to date = '26-08-2018'
;

WITH Items (COATitle) AS (
    SELECT DISTINCT COATitle
    FROM dbo.[Transactions]
), OpeningBalance (COATitle, OpeningAmount) AS (
    SELECT COATitle, SUM(ISNULL(DebitAmount, 0)) - SUM(ISNULL(CreditAmount, 0))
    FROM dbo.[Transactions]
    WHERE TransactionDate < @from
    GROUP BY COATitle
), DebitCredit(COATitle, DebitAmount, CreditAmount)  AS (
    SELECT COATitle, SUM(ISNULL(DebitAmount, 0)), SUM(ISNULL(CreditAmount, 0))
    FROM dbo.[Transactions]
    WHERE (@from <= TransactionDate) AND (TransactionDate <= @to)
    GROUP BY COATitle
)
SELECT 
    i.COATitle,
    OpeningDebitAmount = (CASE WHEN SUM(ISNULL(ob.OpeningAmount, 0)) < 0 THEN 0 ELSE SUM(ISNULL(ob.OpeningAmount, 0)) END),
    OpeningCreditAmount = (CASE WHEN SUM(ISNULL(ob.OpeningAmount, 0)) < 0 THEN -SUM(ISNULL(ob.OpeningAmount, 0)) ELSE 0 END),
    DebitAmount = SUM(ISNULL(dc.DebitAmount, 0)), 
    CreditAmount = SUM(ISNULL(dc.CreditAmount, 0)),
    ClosingDebitAmount = (CASE WHEN SUM(ISNULL(ob.OpeningAmount,0)+ISNULL(dc.DebitAmount,0)-ISNULL(dc.CreditAmount, 0)) < 0 THEN 0 ELSE SUM(ISNULL(ob.OpeningAmount,0)+ISNULL(dc.DebitAmount,0)-ISNULL(dc.CreditAmount, 0)) END),
    ClosingCreditAmount = (CASE WHEN SUM(ISNULL(ob.OpeningAmount,0)+ISNULL(dc.DebitAmount,0)-ISNULL(dc.CreditAmount, 0)) < 0 THEN -SUM(ISNULL(ob.OpeningAmount,0)+ISNULL(dc.DebitAmount,0)-ISNULL(dc.CreditAmount, 0)) ELSE 0 END)
FROM Items i
LEFT JOIN OpeningBalance ob ON (i.COATitle = ob.COATitle)
LEFT JOIN DebitCredit dc ON (i.COATitle = dc.COATitle)
GROUP BY i.COATitle

Here's a possible solution for you. First, sample data:

declare @Transactions table
(
    ID int, 
    COATitle varchar(35),
    ConfigurationCode int, 
    DebitAmount money, 
    CreditAmount money,
    TransactionDate date
);

insert @Transactions values
    (1, 'Sales', 24, 2400, NULL, '20180824'),
    (2, 'Items', 24, NULL, 1200, '20180824'),
    (3, 'Bank', 24, NULL, 1200, '20180824'),
    (4, 'Meezan', 24, 1500, NULL, '20180825'),
    (5, 'Items', 24, NULL, 1500, '20180825'),
    (6, 'Bank', 24, NULL, 1200, '20180826'),
    (7, 'Sales', 24, 5400, NULL, '20180826'),
    (8, 'Items', 24, NULL, 1200, '20180826'),
    (9, 'Bank', 24, NULL, 3000, '20180826');

A couple of things to note here. There's no need to use quotes to delimit numeric literals as you've done in the original question, and I would generally advise against using the numeric data type without specifying the precision and scale explicitly. I've chosen the money type for debit and credit amounts instead.

Next, I'll use a couple of local variables to control the operation of the query. For your first test case we'll use:

declare @BeginDate date = '20180801';
declare @EndDate date = '20180824';

Here's the implementation:

with RawDataCTE as
(
    select
        T.COATitle,
        OpeningDebit = coalesce(sum(case when T.TransactionDate < @BeginDate then T.DebitAmount end), 0),
        OpeningCredit = coalesce(sum(case when T.TransactionDate < @BeginDate then T.CreditAmount end), 0),
        DrAmount = sum(case when T.TransactionDate between @BeginDate and @EndDate then T.DebitAmount end),
        CrAmount = sum(case when T.TransactionDate between @BeginDate and @EndDate then T.CreditAmount end)
    from
        @Transactions T
    group by
        T.COATitle
)
select
    R.COATitle,
    R.OpeningDebit,
    R.OpeningCredit,
    R.DrAmount,
    R.CrAmount,
    ClosingDebit = R.OpeningDebit + coalesce(R.DrAmount, 0),
    ClosingCredit = R.OpeningCredit + coalesce(R.CrAmount, 0)
from
    RawDataCTE R
where
    R.OpeningDebit > 0 or
    R.OpeningCredit > 0 or
    R.DrAmount > 0 or
    R.CrAmount > 0;

The CTE groups everything by COATitle, and as I've assumed from your desired results, produces non-null results for the opening balances but may produce null results for the sum of debits and credits that fall within the desired time frame. The CTE doesn't try to decide which COATitle records should be included or excluded because we need to have the data aggregated before making that decision.

The query outside the CTE builds the closing balances from the CTE data and omits any COATitle for which there are no opening balances and no entries made during the time period (and therefore no closing balances). When I run the query for 2018-08-01 through 2018-08-24, the result is:

And for 2018-08-25 through 2018-08-26, the result set is:

Technically you could do this without a CTE if you wanted; you'd just have to use a HAVING clause instead of a WHERE to determine which accounts are to be included, and you'd have to define ClosingDebit and ClosingCredit independently rather than as sums of the other fields in the result set. For instance, this will work:

select
    T.COATitle,
    OpeningDebit = coalesce(sum(case when T.TransactionDate < @BeginDate then T.DebitAmount end), 0),
    OpeningCredit = coalesce(sum(case when T.TransactionDate < @BeginDate then T.CreditAmount end), 0),
    DrAmount = sum(case when T.TransactionDate between @BeginDate and @EndDate then T.DebitAmount end),
    CrAmount = sum(case when T.TransactionDate between @BeginDate and @EndDate then T.CreditAmount end),
    ClosingDebit = coalesce(sum(case when T.TransactionDate <= @EndDate then T.DebitAmount end), 0),
    ClosingCredit = coalesce(sum(case when T.TransactionDate <= @EndDate then T.CreditAmount end), 0)
from
    @Transactions T
group by
    T.COATitle
having
    sum(case when T.TransactionDate <= @EndDate and (T.DebitAmount > 0 or T.CreditAmount > 0) then 1 else 0 end) > 0;

I find the CTE version a little easier to read and understand, but your mileage may vary.