Fan continuously running without any load
Your problem is essentially a thermal monitoring failure and in all likelihood your CPU and GPU will be working within their thermal limits (in fact they will probably be running cooler than usual thanks to you fan running at 100% PWM). While this can be caused by a failed sensor or SMC it can also be caused by power issues close to the SMBus (which is responsible for simple communication between various hardware components). Take a look at this Louis Rossman video where a very similar issue is diagnosed (spoiler alert, surprisingly it turns out to be a faulty FaceTime camera causing the fan speed issue)...

2014 MacBook thermal monitoring failure with fans running at maximum speed

The precise nature of the fault is largely academic as Apple will simply replace the logic board rather than attempt a component-level repair. An independent such as Rossmann may well be able to diagnose the fault and repair it at a reasonable price, but the only free/DIY option is to install software to bring your fan speeds under control and live with it.

If you want to pay for a repair you will need to use a monitoring utility to give you an idea of whether it's a single or multiple sensor failure. I use HWSensors as it's free, works and gives comprehensive system information, but any of the other suggested utilities should give you the feedback you need. You'll definitely be wanting to check both CPU and GPU temperatures look to be within normal limits at idle, if either is obviously too high or low then the SMC will default to maximum fan speeds to ensure nothing cooks, regardless of whether it's actually cooking or not.

My choice of utility to bring fan speeds under control is Macs Fan Control, this allows you to define your reference sensor and is also well-suited for all iMac models with custom hard disk firmware which causes the system fans to run at full speed when a non-Apple drive is fitted. This is also arguably superior to the standard OS X thermal control in that it lets you set up a thermal plan to suit you rather than having to live with Apple's predefined thermal-acoustic balance. 

If you want a simpler interface then SSD Fan Control may be more up your street.