No, "globbish" filehandles are package variables and never go out of scope. The "best" variant would look like so:

sub start {
    warn "## start";
    my $self = shift;
    open my $fh, '<', '/path/to/some/image' or confess $!;
    $self->header_props(-type=>'your/mimetype');
    local $/;
    return <$fh>;
}
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Though that does of course do nothing to solve your leak.

Makeshifts last the longest.