What are the rates posted on hotel room doors / in the lobby for?

Couple things, one is the very good reason given above.
Second, lets say you are supposed out check out on Friday morning and the hotel is sold out for Friday evening. Well, now that you have not vacated your room, the person that reserved that room for Friday is out of a place to stay.
Since the hotel has guaranteed them a room reservation they must “walk” them to another hotel and pay for their room, meaning that person pays $0 for their stay.
Meanwhile you are in that room, so you will essentially be charged that maximum rate to accommodate the fact that you stayed in the room and the hotel had to pay to relocate the other guest. So you are essentially paying for 2 rooms.

Some localities call it a civil code, but I think it is uniform commercial code, where I live, being the rooms are rented for purposes of commerce.
Most states adopt some form of commercial or civil code to avoid price gouging, and one requirement is a maximum daily rate shall be posted in a conspicuous location, AND that rate shall be set at a prescribed schedule based upon the services & amenities offered by the hotel, so a Motel 6 is not going to charge like the Ritz Carlton.
Not posting these rates should also be a violation to the civil/commercial code, and at that point the fee would be assumed to be ZERO dollars, plus fines to the local governing body having authority over innkeeper’s, for not posting the rates.
Some states may not control this, but they are in the minority.

Gas stations are also regulated against price gouging.
I have successfully shut down a few businesses for violating state UC codes. Others have paid steep fines.
These posted rates are the maximum rate the hotel can charge for one night, during a special event, or during peak season.

Recently a dive hotel like an old run down Day’s Inn attempted to charge $1000 for one night, but the maximum allowable was $400, for a room meeting those standards.
The room was not posted, so the customer was refunded $3000 + 3x travel and legal expenses, and the owner was fined $10,000, for not posting their rates.

With the solar eclipse looming, price gouging is rampant in some smaller towns.
I put 25 gallons of gasoline in a truck that had a 20 gallon fuel tank. The owner of that station was shut down, and fined.
I spent a lot of time traveling for work, and crooks are quite plentiful. I have seen overcharges on my credit cards more than once, too.

I’ve worked in the hospitality industry for 18 years in the United States so I can answer this with some knowledge. Most states require a hotel to post their “Maximum Possible Rate” on the back of the guest room door. This is to keep a hotel from “price gouging” in the event of a natural disaster or even from discriminatory practices, (charging someone of a different race a ridiculous rate). While these rates may seem high to the average traveler, you must keep in mind, there may be a Presidential Suite or some other luxury accommodations within the hotel that is already offered at a much higher rate and the law usually states there must not be different rates posted in different rooms. Also, most hotels charge a much higher than “standard” rate during special events in the city or a certain time of year. And finally, because these documents are generally produced once every 10 years, the hotel is trying to account for inflation and just about any other reason they can think of that would cause a rate increase. So, while the rate may seem insanely high right now, in 10 years, that could be the new, New Year’s Eve rate. The reason a hotel couldn’t simply list, “1 million dollars” is because the law also requires the rate to be “somewhat reasonable”. I hope this helps anyone who’s traveling and has this same question in the United States!

You may actually be asked to pay this amount if you ask for a room at the reception. It happened to me once in Hong-Kong. We had 2 nights booked over the internet, and wanted to extend our stay. The manager at the reception advised us to go to an internet cafe round the corner and do the booking there…

California, USA is one place where you see these posted rates.

I believe this is to comply with Civil Code Section 1863:

(a) Every keeper of a hotel, inn, boardinghouse or
lodginghouse, shall post in a conspicuous place in the office or
public room, and in every bedroom of said hotel, boardinghouse, inn,
or lodginghouse, a printed copy of this section, and a statement of
rate or range of rates by the day for lodging.

(b) No charge or sum shall be collected or received for any
greater sum than is specified in subdivision (a). For any violation
of this subdivision, the offender shall forfeit to the injured party
one hundred dollars ($100) or three times the amount of the sum
charged in excess of what he is entitled to, whichever is greater.
There shall be no forfeiture under this subdivision unless notice be
given of the overcharge to such keeper within 30 days after payment
of such charges and such keeper shall fail or refuse to make proper
adjustment of such overcharge.

I assume the intention was to prevent things like price gouging for distressed travelers, special events, or customers the management sees as undesirable (e.g. racism). However, as you point out, hotels often set this rate very high, so that they can charge high rates when they want, and claim everyone else is getting a discount.

Given this, I’m not sure why they stop at prices that are 3-5 times the actual rates; why not post a rate of a million dollars per night, and have complete freedom? There may be some additional regulations that I don’t know about.