What is the effect on a 2014 MacBook Pro 15" when the internal fans stopped working

Short answer: it won't work for long.

Longer answer: macOS will resort to a number of increasingly desperate measures to keep the Mac's internal temperatures within limits. If they are unable to do that, a separate (hardware-based) system will unceremoniously shut down the Mac on its own.

A MBP's internal fans are easily replaced and at a reasonable cost. Start here: Official Apple Support. They might simply be obstructed with dust or other contaminants, preventing them from working.

Try an SMC reset before doing anything else.

Reset the SMC of your Mac - Apple Support

Has any work been done on this computer? A 15-inch Macbook Pro has two fans and it would be rare for both to fail unless some work inside damaged a sensor or some connector.

Have you run diagnostics?

You can check but I do not believe Apple will work on a 2014 model any longer. You may need to look for an independent provider if you cannot fix it yourself.

The fans themselves are very inexpensive. A few bucks. Installation will be extra. Start here: Official Apple Support. Apple is likely to refer you to an authorized Apple repair facility, if so use Explore Service Locations.

As I wrote an obstruction or dust might be interfering with fan operation. Apple Diagnostics will exercise the fans and let you know if they aren't working properly: Use Apple Diagnostics to test your Mac - Apple Support

Please bear in mind this site is awash with reports of people attempting to repair their own Macs that resulted in sparks and smoke and a dead Mac, asking us what to do now. Don't be one of them.

Opposite opinion here based on working on several newer Macbook Pros: I say it won't help much.

On the older "silver-key" Macbook Pros (pre about 2009) the case bottom touched the heat sink and was part of the heat dissipation system. A "chill pad" worked really well on those.

In current Macbook Pro form factors, the logic board hangs from the keyboard deck and does not touch the bottom plate. A chill pad is far less effective.

Part of that is due to geometry. In most current Macbook Pros each fan blows directly on a radiator-type heat exchanger:

Ignore the yellow arrows.

The heat exchanger is the finned object at the edge of the fan recess. The fan housing actually touches the exchanger to ensure max cool air delivery.

You must have a strong flow of air directly across this device or risk damage, possibly irreparable.

I cannot see how a chill pad can service that need.

At the bottom of my mid 2014 (A2398) MBP are 3 vent on each side...

Yes, I know, but I fear the air moving outward across the bottom the case will actually reduce the air volume or even stop cool air entering those vents due to a Venturi or Bernoulli effect:

I believe your best non-Apple cooling option given your current circumstances and needs is a small fan on each side of the case blowing directly into the vents. At least you know it is going in the right direction that way.

I cannot see how a chill pad can service that need.

Agreed.

I think I paid $6 for a replacement fan. That might have even been for a L / R pair. But if the OP wants to spend $20-$30 on a solution unlikely to work, fine.

Sure, try it.

Were you able to confirm the internal fans are not working properly?

Use Apple Diagnostics to test your Mac - Apple Support

Hope is not a valid engineering principle.