Yes and no. Yes there can be some logic behind it for GME (and other cases specifically), but no generally the logic is “flawed”.
First let’s remember what shorting a stock means: You borrow a stock from someone (usually a broker), for an agreed period of time. When that period is up, you give the stock back. In between, you sell that stock in the market and hope for the price to drop, and then you buy it back again before you need to deliver it back to its rightful owner. Essentially you sell something that you don’t own, hence you are one stock short.
The benefit of this is the opposite of buying stocks, where you hope for the price to go up; here you hope (expect) the prices to go down.
So what happens if the price doesn’t go down before you need to deliver the stock back? You loose money (potentially infinite if the price explodes).
To answer the “no” part first: generally holding the stocks will not force the prices up when it is time for the short positions to be cashed in, because they generally don’t make up a huge chunk of the free floating stocks - so the investors will just slowly pick up the stocks in the market without really affecting the price (and few retail investors holding onto their stocks are usually insignificant volumes, nothing that affect prices at all).
To the yes part: What makes the GME case different? The amount of short positions compared to the free floating stocks are huge. Many institutional investors are betting against GME (arguably rightfully so), so much that if all short positions were to be cashed in on the same day, there wouldn’t be enough free stocks floating; essentially resulting in what is called a short squeeze; I.e. investors need to deliver the stock back, and because they don’t own it, they need to go buy it in the market regardless of the price - which will drive up the price.
Now, everyone doesn’t need to deliver the stocks on the same day, because each contract is different, and no one (except for the involved parties) really know about the contractual terms - you can only speculate and come up with qualified guesses. But if no one is selling any of the stock, the prices will go up once the short positions need to be cashed in (well in reality prices can only go up if there is a trade, so someone are for sure selling, but if the majority won’t sell, the prices go up).
The expectation around GME (and AMC, NAKD, NOK etc.) is that the short positions need to be cashed in soon; and the r/wallstreetbets community have been trying to get this message out to as many people as possible, encouraging people to buy as much as possible (this drives up prices) and just holding it (I.e not selling!); the idea is that the institutional investors cannot cash in slowly at market rate, but will be forced to pay an extreme premium which will result in a short squeeze where prices are driven up to the extreme.
The narrative or justification for this is that the short-sellers are the ones driving down the prices; and people somehow think they are saving GameStop by doing this (they don’t).
If this happens, and people get out on top, they stand to make a serious return on their investments; however if you do not get out on top you stand to take a huge dive (in time all the way back to the origin, which was very low, causing the people who didn’t get out to lose upwards of 100% of their investment).
Long story, but hopefully this shed some light on the situation.