Lightroom Eating Up Memory & Overheating Brand New Macbook Pro

"and the computer heats up to the point where the fans turn on and run very loudly."

The Macbook is not "overheating" when it runs the fans. 

LR is designed to use as much of the CPU (processors) as it can, as well as the GPU (the graphics processor), and it uses much more CPU and GPU than the typical app (e.g. email, a Web browser, or video player).  LR's utilization of CPU/GPU is often many times the utilization of other apps, even when doing seemingly innocuous things like scrolling Library grid view or moving sliders in Develop.  This consumes a lot of power, which must be dissipated as heat.  

Macbooks are notorious for running the fans whenever an app like LR starts using the CPU and GPU.   Apple likes to make their laptops as slim as possible while still advertising the most powerful CPUs and GPUs, which means cramming a lot of power-hungry chips into very small spaces, which means turning on the fans as soon as resource-intensive apps start actually using the CPU and GPU to any degree.

See this article for some tips that might reduce when the fans run: https://support.apple.com/en-us/HT202179

I've owned fully loaded Macbook Pros for seven years, and their fan noise drives my wife bonkers.

As you can see from the info in my signature, I too, have a MACBook Pro, this a mid 2015 Retina 15".

 I use the MACBook when on a Photon Trip (a.k.a. vacation) . And when I do use it, to import into a catalog, color grade the days shoots, and to do some post processing to see how I am doing, general lightroom work. I do not have these issues.

But some questions about your experience.

1. With or without the notebook plugged into electrical power?

2. Catalog on internal hard drive, or external?

3. Less important, photos on internal or external?

4. In my case i did not have these issues you are having on previous OS, or before the supplemental update. But, what is your OS version. If v10.14.6, have you installed the supplemental update

https://www.macworld.com/article/3433951/apple-releases-macos-mojave-10-14-6-supplemental-update.html

5. How much room is on your startup disk?

and see the following link, item 5 above comes from it, and plenty others to study.

http://www.thexlab.com/faqs/performance.html

How many apps do you have  open at one time, how large  is your catalog? If you create a test  catalog with a handful of  images in it, does it still overheat? When Lightroom is  open have you checked your Activity Monitor to check application  utilization? In preferences do you have Use GPU selected? In  Preferences> Performance what is your Camera Raw Cache Settings set to? Have you tried to optimize your  catalog? A brand new computer with 16 GB of RAM shouldn't be overheating.