What would happen to the price of a high-demand, no-supply asset once liquidity reaches 0?

There is no trading price if there is no trade. There would probably be quotes at a higher and higher bid price by potential buyers until someone decides to sell.

You are misunderstanding how pricing works on a stock.

The price you see quoted as the stock price is the last transaction that actually took place. That is, a buyer and a seller agreed on that price. If it gets to a point where no seller wants to give up stock at a price that any buyer is willing to pay, then no sales take place and the quoted stock value does not change.

In addition to the quoted stock price, there is a bid price and an ask price. The bid price is the highest price that buyers are currently offering to buy the stock at. The ask price is the lowest price that sellers are currently offering to sell at. At anytime, a buyer could come along and offer to buy at the ask price; a sale would take place, and the quoted stock value would update and increase. Alternatively, at any time a seller could decide to sell at the bid price, at which point the quoted stock value would decrease.

If there was an asset with a fixed supply and ever increasing demand, what would happen to its price once all holders of said asset refused to sell? Obviously this is practically impossible, but just hypothetically?

I don't think it's impossible at all. It happens on a regular basis.

One example (but maybe a silly example) is elixir of eternal youth. There are undoubtedly thousands if not millions of people who would pay enormous amounts of money for an elixir of eternal youth. However, the supply is zero, because no elixir of eternal youth exists.

Another known example, perhaps, is 1933 double eagle coins. Only 13 of them are known to exist, and none of them are for sale. However, the demand for these coins is undoubtedly very high, and may be rising all of the time.

So, what is happening to the price of a 1933 double eagle coin right now? Well, it depends on exactly what you mean by "the price." There is no one particular number which is called "the price" of an asset.

You could interpret "the price" as meaning the last price that one of the coins was sold at. That price is $7,590,020, and that price is going to remain at $7,590,020 until one of the coins is sold again.

You could interpret "the price" as meaning the smallest price that someone is willing to sell one of the coins at. However, it appears that nobody is willing to sell any of the coins at any price, so if you define "the price" this way, then the coins have no price.

You could interpret "the price" as meaning the largest price that someone is willing to buy one of the coins at. However, this number is difficult to determine, because there is currently nobody who's keeping track of that information for 1933 double eagles.

To answer your last questions:

What if nobody ever [sold]?

Then nobody would ever be able to buy.

How could something even have a price with zero liquidity?

As above, it depends on what you mean by "price."