The background colour works fine in principle, the problem is that highlighting macro you use fill not only lines to highlight, it fill all other lines with white and thus covers up the gray background. 

As a workaround, you can adjust the `\btLstHL` as in the following example:

    \documentclass{beamer}
    
    \usepackage{listings, pgffor}
    
    \makeatletter
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %
      % \btIfInRange{number}{range list}{TRUE}{FALSE}
      %
      % Test if int number <number> is element of a (comma separated) list of ranges
      % (such as: {1,3-5,7,10-12,14}) and processes <TRUE> or <FALSE> respectively
    
    \newcount\bt@rangea
    \newcount\bt@rangeb
    
    \newcommand\btIfInRange[2]{%
        \global\let\bt@inrange\@secondoftwo%
        \edef\bt@rangelist{#2}%
        \foreach \range in \bt@rangelist {%
            \afterassignment\bt@getrangeb%
            \bt@rangea=0\range\relax%
            \pgfmathtruncatemacro\result{ ( #1 >= \bt@rangea) && (#1 <= \bt@rangeb) }%
            \ifnum\result=1\relax%
                \breakforeach%
                \global\let\bt@inrange\@firstoftwo%
            \fi%
        }%
        \bt@inrange%
    }%
    \newcommand\bt@getrangeb{%
        \@ifnextchar\relax%
            {\bt@rangeb=\bt@rangea}%
            {\@getrangeb}%
    }%
    \def\@getrangeb-#1\relax{%
        \ifx\relax#1\relax%
            \bt@rangeb=100000%   \maxdimen is too large for pgfmath
        \else%
            \bt@rangeb=#1\relax%
        \fi%
    }%
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %
      % \btLstHL<overlay spec>{range list}
      %
    \newcommand<>{\btLstHL}[1]{%
        \only#2{\btIfInRange{\value{lstnumber}}{#1}{\color{blue!30}}{\color{gray!20}}}%
    }%
    
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      %
      % \btInputEmph<overlay spec>[listing options]{range list}{file name}
      %
    \newcommand<>{\btLstInputEmph}[3][\empty]{%                                                    
        \only#4{%
            \lstset{linebackgroundcolor=\btLstHL{#2}}%
            \lstinputlisting{#3}%
        }% \only
    }%
    \makeatletter
        \let\old@lstKV@SwitchCases\lstKV@SwitchCases
        \def\lstKV@SwitchCases#1#2#3{}
    \makeatother
    \usepackage{lstlinebgrd}
    \makeatletter
        \let\lstKV@SwitchCases\old@lstKV@SwitchCases
        \lst@Key{numbers}{none}{%
        \def\lst@PlaceNumber{\lst@linebgrd}%
        \lstKV@SwitchCases{#1}%
            {none:\\%
            left:\def\lst@PlaceNumber{\llap{\normalfont
                \lst@numberstyle{\thelstnumber}\kern\lst@numbersep} \lst@linebgrd}\\%
            right:\def\lst@PlaceNumber{\rlap{\normalfont
                \kern\linewidth \kern\lst@numbersep
                \lst@numberstyle{\thelstnumber}}\lst@linebgrd}%
        }{\PackageError{Listings}{Numbers #1 unknown}\@ehc}}
    \makeatother
    
    \begin{document}
    
    \begin{frame}[fragile]{Problem}
    Test programme:
    \begin{lstlisting}[%
        language=C, 
        gobble=4, 
       linebackgroundcolor={%
           \btLstHL<1>{1,3,5-6}%
           \btLstHL<2>{4}%
       },
        backgroundcolor=\color{gray!20}
    ]
        #include <stdio.h>
    
        int main(void) {%
        printf("Hello World!");  
            return 0;
        }
    \end{lstlisting}
    \end{frame}
    \end{document}

[![enter image description here][1]][1]


  [1]: https://i.stack.imgur.com/hRKPb.png