I recently experienced and resolved this issue on a 13-inch mid 2012 MacBook Pro, so I'm sharing the steps I took for the sake of posterity.
=== Background ===
I have a 2012 MacBook Pro running a clean install of Linux. Right after installation, I noticed the laptop was prone to speed throttling and overheating.
=== Troubleshooting ===
My first step was to run [link|https://support.apple.com/en-gb/102550|Apple Diagnostics|new_window=true] by holding the D key on startup. This returned error codes PPF003 and PPF004, which according to Apple indicate a fault with the fan.
I also used a third party tool to manually control the fan from Linux, and this reported the fan was intermittently spinning up and not cooling the laptop sufficiently. The temperature sensors would quickly reach 105°C, causing the issues with overheating.
=== Solution ===
First, I opened up the rear of the laptop and cleaned the accumulated dust, particularly around the cooling vents.
I then took apart the fan, cleaned it and applied a drop of sewing machine lubricant to the shaft of the electric motor.
After reassembly, I ran Apple Diagnostics again, and this time it did not report any faults related to the fan.
The fan now spins at normal speeds and the third party tool in Linux reports temperatures of between 70°C and 76°C during regular use with no throttling of performance.
This was originally intended as a temporary fix while I planned to source for a replacement fan. However, after a month of regular use, I feel confident enough in the fix to share my solution.