Loop in LaTeX is not working

Your code worked for me with some minimal files, but \inputting files in the middle of a loop might go wrong, as Phelype pointed out in his comment.

I'd use a somewhat safer approach by defining a macro in the loop which collects the various \input and calling this macro when the loop is done. I've changed your definition such that \MakeLecture{1}{4} inserts the files from 1 to 4 (and not 3 as in your case) because it sounds more reasonable to me but you can just delete the line \advance\count@\m@ne if you don't want that.

\documentclass{article}

\makeatletter
\newcommand{\MakeLecture}[2]{%
   \count@=#1
   \advance\count@\m@ne
   \def\@inputmyfiles{}%
   \loop\ifnum\count@<#2
      \advance\count@\@ne
      \edef\@inputmyfiles{\unexpanded\expandafter{\@inputmyfiles}\noexpand\input{Chapter0\the\count@}}%
   \repeat
   \show\@inputmyfiles % ---> just for testing, delete this in real use
   \@inputmyfiles
}
\makeatother

\begin{document}
\MakeLecture{2}{5}
\end{document}

The \show command returns

> \@inputmyfiles=macro:
->\input {Chapter02}\input {Chapter03}\input {Chapter04}\input {Chapter05}.

A one-liner with expl3 (can be nested):

\RequirePackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand \MakeLecture { O{1} m }
  { \int_step_inline:nnn {#1} {#2} { \input {Chapter0##1} } }
  % { \int_step_inline:nnn {#1} {#2} { \input {\code/Lecture~Slides/Chapter0##1} } }
\ExplSyntaxOff

\MakeLecture{4} % same as \MakeLecture[1]{4}