As far as i can tell, exec is not the right tool. For a more controlled approach, you may use proc_open. This may look something like this:

$process = proc_open(
   'ssh -f -L 3400:127.0.0.1:3306 [email protected] sleep 1',
   [/*stdin*/ 0 => ["pipe", "r"], /*stdout*/ 1 => ["pipe", "w"], /*stderr*/2 => ["pipe", "w"]],
   $pipes
);

// Set the streams to non-blocking
// This is required since any unread output on the pipes may result in the process still marked as running
// Note that this does not work on windows due to restrictions in the windows api (https://bugs.php.net/bug.php?id=47918)
stream_set_blocking($pipes[1], 0);
stream_set_blocking($pipes[2], 0);

// Wait a litte bit - you would probably have to loop here and check regulary
// Also note that you may need to read from stdout and stderr from time to time to allow the process to finish
sleep(2);

// The process should now be running as background task
// You can check if the process has finished like this
if (
    !proc_get_status($process)["running"] ||
    proc_get_status($process)["signaled"] ||
    proc_get_status($process)["stopped"] ||
) {
   // Process should have stopped - read the output
   $stdout = stream_get_contents($pipes[1]) ?: "";
   $stderr = stream_get_contents($pipes[2]) ?: "";

   // Close everything
   @fclose($pipes[1]);
   @fclose($pipes[2]);
   proc_close($process);
}

You can find more details on that the manual on proc_open

If I add 2>&1 to my command the error output just goes nowhere since stdout is already being piped to /dev/null.

You can redirect stdout to null and stderr to stdout. That would seem to me as the simpler way of doing what you want (minimal modification).

So instead of

>temp.log 2>&1

do:

2>&1 1>/dev/null

Note that the order of the redirects is important.

Test

First we exec without redirection, then we redirect as above to capture stderr.

<?php
    $me = $argv[0];

    $out = exec("ls -la no-such-file {$me}");

    print("The output is '{$out}'\n");

    print("\n----\n");

    $out = exec("ls -la no-such-file {$me} 2>&1 1>/dev/null");

    print("The output is '{$out}'\n");

    print("\n");
~

$ php -q tmp.php

ls: cannot access 'no-such-file': No such file or directory
The output is 'The output is '-rw-r--r-- 1 lserni users 265 Oct 25 22:48 tmp.php'

----
The output is 'ls: cannot access 'no-such-file': No such file or directory'

Update

This requirement was not clear initially: "process must detach" (as if it went into the background). Now, the fact is, whatever redirection you do to the original stream via exec() will prevent the process from detaching, because at the time the detachment would happen, the process has not completed, its output is not delivered.

That is also why exec() reports a zero error code - there was no error in spawning. If you want the final result, someone must wait for the process to finalize. So, you have to redirect locally (that way it will be the local file that will wait), then reconnect with whoever it was that waited for the process to finalize and then read the results.

For what you want, exec will never work. You ought to use the proc_* functions.

You might however force detach even so using nohup (you have no control over the spawned pid, so this is less than optimal)

if (file_exists('nohup.out')) { unlink('nohup.out'); }
$out = shell_exec('nohup ssh ... 2>&1 1>/dev/null &');
...still have to wait for connection to be established...
...read nohup.out to verify...
...
...do your thing...

As I said, this is less than optimal. Using proc_*, while undoubtedly more complicated, would allow you to start the ssh connection in tunnel mode without a terminal, and terminate it as soon as you don't need it anymore.

Actually, however, no offense intended, but this is a "X-Y problem". What you want to do is open a SSH tunnel for MySQL. So I'd look into doing just that.