In your TestEK.txt file, you have all  {[%tqu ... ]} sequences on the same line. In the second, real file, these sequences span two or more lines.

You are processing the file line-by-line and matching your regex against each line, so if a  {[%tqu ... ]} sequence spans multiple lines, the regex will never see it.

If the file is small, less than, say, several hundred megabytes and never likely to grow larger, it might be easiest to "slurp" the entire file at once as a string into a scalar variable and then do a single  s/// against the variable, then write the string back out to the new file.
    my $string = do { local $/;  <>; };
    $string =~ s/$regex/$subst/gis;
    print $string;
(untested). Note that the  s/// now needs a  /s regex modifier so that  . (dot) in  .* will match a newline across multiple lines. Get rid of the while-loop entirely.

Update: See also File::Slurp.

c:\@Work\Perl\monks\OldChamp>perl -wMstrict -le
"my $s = do { local $/;  <>; };
 print qq{[[$s]] \n};
 ;;
 my $rx = '{\[%tqu.*]}';
 my $su = '';
 $s =~ s/$rx/$su/gis;
 print qq{[[$s]]};
"  tqu.txt
[[keep this {[%tqu get
rid of
this]} and
keep this too
]]

[[keep this  and
keep this too
]]
[download]

Update 3: Update 2 contains a rookie mistake: using greedy  .* instead of the lazy  .*? version. Here's a version that will actually work with a single long string. The previous version would delete everything between the absolute first  {[%tqu and the absolute last  ]} sequence in the file.

c:\@Work\Perl\monks\OldChamp>perl -wMstrict -le
"my $s = do { local $/;  <>; };
 print qq{[[$s]] \n};
 ;;
 my $rx = '{\[%tqu.*?]}';
 my $su = '';
 $s =~ s/$rx/$su/gis;
 print qq{[[$s]]};
"  tqu.txt
[[keep this {[%tqu get
rid of
this]} and
keep this too
keep {[%tqu but
dump
also ]} it
to here.
]]

[[keep this  and
keep this too
keep  it
to here.
]]
[download]