On startup, I get the following error messages every time:

“To interrupt normal setup press enter.” Then it beeps and I get the message “fan error. Press ESC  to startup with limited performance. If I don’t hit ESC,  it just shuts off. If I do press ESC, the fan does not run at all. 

Hello guest2472,

Welcome to Lenovo Community

I understand that you're having fan error on start up with your ThinkPad.
To start, remove all external peripherals.
Reload the BIOS default settings.
Update the BIOS, drivers and firmware thru the support site.
Test and observe.
If issue persists, run diagnostics.
Hope this helps.

Regards, 
Lorenzo_Lenovo

BIOS says it’s up to date. I’ve run diagnostics and the fan failed. What should I do next?

The error means the fan is running too slowly or not at all.  In most cases, the fix is simply to replace the fan,

Hi Jcarlin76,

How's the system doing?

If you would like to look into repairing the system yourself, maybe some DIY? You can find part numbers for your system after entering the serial or type number on this site
Parts lookup: https://support.lenovo.com/partslookup?cid=all:apps:hxatlt

General – how to find and buy parts & accessories
https://support.lenovo.com/solutions/ht036683-how-to-look-up-and-purchase-accessories-upgrades-and-frucru-replacement-parts

For accessories: https://smartfind.lenovo.com/#/country-language

You may also reach out to the tech support or nearest Lenovo Service Center.
Cheers 

Regards, 
Lorenzo_Lenovo

You can do two things that have worked for me in the past.

1.  get a can of compressed air and direct it into the exhaust fan port.  you can get a lot of dust out and you can hear the fan spinning.

2.  more recently, the above didn't work for me so i got a hair dryer,  set it to high or low (depending on how cold where you are) and let it heat up the fan     exhaust port about 30 seconds, then turn on the laptop's power while still running the hair dryer.  WORKED!!!!

Hello limelight50,
Welcome to Lenovo Community
We appreciate for expressing your suggestions.
This is another step that the original poster can try.

Regards, 
Lorenzo_Lenovo