Escape single quote in openquery using dynamic query

You need single quotes around your variables since you are trying to make them string literals. But also complicating it is the fact that you are trying to create a SQL statement in a string that includes another SQL statement in a string. So you need to make your line read like:

And cases.code IN (''''' + @A +''''', ''''' + @B + ''''')

You need two sets of double quotes so that the string literal inside of your string literal is interpreted correctly. Huh? Right. :)

Ultimately you need to build a string that has this valid SQL Syntax in it:

Select * From Openquery(LinkedServerName,'SELECT cases.casenum, user.username, code
            From cases
            Inner join user
            On cases.casenum = user.user_id
            Where cases.date_opened > DateAdd(day,1-datepart(dw,Convert(date,20150501)), Convert(date,20150501))
            And cases.date_opened <= DateAdd(day,8-datepart(dw,Convert(date,20150501)), Convert(date,20150501))
            And cases.code IN (''AAA'', ''BBB'')
            ORDER BY casenum')

You need two quotes around the AAA and BBB in your inner SQL string because it is also SQL code inside a string. So you need double double quotes to get double quotes inside your main string you are building.

Using Brian Pressler response i suggest to create a function to format the params if you have various openquery calls, sometimes quoting a lot could be eyebreaker.

CREATE FUNCTION [dbo].[Ufn_QuoteFormat]   
( @param varchar(200) --Modify accord your requirement)     
RETURNS varchar(208)
AS    
BEGIN   
  DECLARE @SingleQuote char(1) = CHAR(39)   
  RETURN  @SingleQuote + @SingleQuote + @param + @SingleQuote +@SingleQuote   
END

Then you can use it in your openqueries like this:

Declare @OPENQUERY nvarchar(500), @TSQL nvarchar(max), @LinkedServer nvarchar(20), @A varchar(5), @B varchar(5), @PickedDate varchar(8)
Set @PickedDate = '20150501'
Set @A = 'AAA'
Set @B = 'BBB'
Set @LinkedServer = 'LinkedServerName'
Set @OPENQUERY = 'Select * From Openquery('+ @LinkedServer + ','''
Set @TSQL = 'SELECT cases.casenum, user.username, code
            From cases
            Inner join user
            On cases.casenum = user.user_id
            Where cases.date_opened > DateAdd(day,1-datepart(dw,Convert(date,' + @PickedDate + ')), Convert(date,' + @PickedDate + '))
            And cases.date_opened <= DateAdd(day,8-datepart(dw,Convert(date,' + @PickedDate + ')), Convert(date,' + @PickedDate + '))
            And cases.code IN (' + [dbo].[Ufn_QuoteFormat](@A) +', ' + [dbo].[Ufn_QuoteFormat](@B) + ')
            ORDER BY casenum'')'
Exec (@Openquery+@TSQL)