tee: /proc/acpi/ibm/fan: Invalid argument

The problem is not with your tee command - although since you are already in a root shell you could have used a simple redirection echo disable > /proc/acpi/ibm/fan (and could certainly have removed the sudo).

Although "everything in Linux is a file", the reality is that "files" in /proc are really interfaces to the running kernel and don't behave exactly like regular files - if we use strace we can see that the error doesn't really come from tee, but from the underlying write call to the open file descriptor (I changed the syntax from a pipe to a here-string to make sure that we are tracing the tee not the echo):

root@t400s:~# strace tee /proc/acpi/ibm/fan <<< 'disabled'
execve("/usr/bin/tee", ["tee", "/proc/acpi/ibm/fan"], [/* 23 vars */]) = 0
.
.
.                              = 0
openat(AT_FDCWD, "/proc/acpi/ibm/fan", O_WRONLY|O_CREAT|O_TRUNC, 0666) = 3
read(0, "disabled\n", 8192)             = 9
write(1, "disabled\n", 9disabled
)               = 9
write(3, "disabled\n", 9)               = -1 EINVAL (Invalid argument)
write(2, "tee: ", 5tee: )                    = 5
write(2, "/proc/acpi/ibm/fan", 18/proc/acpi/ibm/fan)      = 18
.
.
.
+++ exited with 1 +++
root@t400s:~# 

The reason the argument is invalid is that by default, the thinkpad_acpi module does not enable fan control - if you cat the file, you will likely see just a few lines as follows:

$ cat /proc/acpi/ibm/fan
status:     enabled
speed:      1838
level:      auto

In order to enable fan control, you need to pass an option to the module during loading, which you can do using a configuration file:

$ cat /etc/modprobe.d/thinkpad_acpi.conf 
options thinkpad_acpi fan_control=1

which you can either create in a text editor, or using

echo 'options thinkpad_acpi fan_control=1' | sudo tee -a /etc/modprobe.d/thinkpad_acpi.conf

(this is just a regular file, so no issues with tee this time). Then you will need to reboot - unfortunately it does not seem to be possible to remove and re-load the thinkpad_acpi module on the fly.

After rebooting to re-load the module with its new option, you should see that it now lists some additional fan_control commands:

$ cat /proc/acpi/ibm/fan
status:     enabled
speed:      3737
level:      auto
commands:   level <level> (<level> is 0-7, auto, disengaged, full-speed)
commands:   enable, disable
commands:   watchdog <timeout> (<timeout> is 0 (off), 1-120 (seconds))

Now you should be able to enable/disable/modify the fan parameters, either by writing to /proc/acpi/ibm/fan directly with a command such as your tee, or using the thinkfan utility described in your link.

For further background about /proc, see What happens when I run the command cat /proc/cpuinfo?