You could do:

if { cmd 2>&1 >&3 3>&- | grep '^' >&2; } 3>&1; then
  echo there was some output on stderr
fi

Or to preserve the original exit status if it was non-zero:

fail_if_stderr() (
  rc=$({
    ("$@" 2>&1 >&3 3>&- 4>&-; echo "$?" >&4) |
    grep '^' >&2 3>&- 4>&-
  } 4>&1)
  err=$?
  [ "$rc" -eq 0 ] || exit "$rc"
  [ "$err" -ne 0 ] || exit 125
) 3>&1

Using exit code 125 for the cases where the command returns with a 0 exit status but produced some error output.

To be used as:

fail_if_stderr cmd its args || echo "Failed with $?"

You could do (POSIXly):

if { cmd 2>&1 >&3 3>&- | grep '^' >&2; } 3>&1; then
  echo there was some output on stderr
fi

Or to preserve the original exit status if it was non-zero:

fail_if_stderr() (
  rc=$({
    ("$@" 2>&1 >&3 3>&- 4>&-; echo "$?" >&4) |
    grep '^' >&2 3>&- 4>&-
  } 4>&1)
  err=$?
  [ "$rc" -eq 0 ] || exit "$rc"
  [ "$err" -ne 0 ] || exit 125
) 3>&1

Using exit code 125 for the cases where the command returns with a 0 exit status but produced some error output.

To be used as:

fail_if_stderr cmd its args || echo "Failed with $?"

You could do:

if { cmd 2>&1 >&3 3>&- | grep '^' >&2; } 3>&1; then
  echo there was some output on stderr
fi

You could do:

if { cmd 2>&1 >&3 3>&- | grep '^' >&2; } 3>&1; then
  echo there was some output on stderr
fi

Or to preserve the original exit status if it was non-zero:

fail_if_stderr() (
  rc=$({
    ("$@" 2>&1 >&3 3>&- 4>&-; echo "$?" >&4) |
    grep '^' >&2 3>&- 4>&-
  } 4>&1)
  err=$?
  [ "$rc" -eq 0 ] || exit "$rc"
  [ "$err" -ne 0 ] || exit 125
) 3>&1

Using exit code 125 for the cases where the command returns with a 0 exit status but produced some error output.

To be used as:

fail_if_stderr cmd its args || echo "Failed with $?"