Perl supports opening a pipe to or from a shell command, that's what the open call seems to try to accomplish. But for that, a pipe sign must be prepended (in case of writing to) or appended (in case of reading from) to the shell expression, like so:

open F, "| some_shell_expression"; # open for writing
open F, "some_shell_expression |"; # open for reading
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See open for details. Your open as written won't work, since you don't specify the open mode (read or write). The command itself as posted by you features a useless use of cat since it is better written as

/usr/bin/mycommand < file
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The next line saves the input record separator (see perlvar), sets it to the empty string enabling "slurp mode" on the filehandle F; next all that can be read from F is read and then passed to eval to be executed. For that to work,

1. something must be read from the filehandle F and
2. the stuff read must be valid perl code.

I suspect that there's nothing read from F, and evaling an empty string doesn't result in an error stored in $@.

If you just want to run a command and check it for errors, use system. If system returns something different than 0 (zero), something went wrong. Then exploit $? (see perlvar), along the lines

    my $cmd = "/usr/bin/mycommand < file";
    if ( system $cmd ) {
        warn $cmd .
          ($? == -1)
            ? " failed to execute: $!\n"
            : ($? & 127)
              ? sprintf "child died with signal %d, %s coredump\n",
                ($? & 127),
                ($? & 128)
                  ? 'with'
                  : 'without'
              : sprintf "child exited with value %d\n", $? >> 8;
    }
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