foreach $file (@ARGV) {
    process($file, 'fh00');
}

sub process {
    my($filename, $input) = @_;
    $input++;               # this is a string increment
    unless (open($input, $filename)) {
      print STDERR "Can't open $filename: $!\n";
      return;
    }

    local $_;
    while (<$input>) {              # note use of indirection
       if (/^#include "(.*)"/) {
           process($1, $input);
           next;
       }
       #...                # whatever
    }
}
[download]

Did you try that under "use strict"?

Can't use string ("fh01") as a symbol ref while "strict refs" in use

From perldoc open:

If FILEHANDLE is an undefined scalar variable (or array or hash element) the variable is assigned a reference to a new anonymous filehandle, otherwise if FILEHANDLE is an expression, its value is used as the name of the real filehandle wanted. (This is considered a symbolic reference, so use strict 'refs' should not be in effect.)

So what you're trying only works with "use strict" if the variable that you use for the filehandle contains an undefined value.