No other answer has mentioned this so I created an account just to say this.
Some membrane proteins do not move. This is because they are fixed in that position in the membrane due to the cytoskeleton. Erythrocytes are a good example of this.
The main protein that is immobilised in erythrocyte membrane is Band 4.1 protein, and its immobilised by Spectrin.
Spectrin forms a tetramer(2 dimers together) that acts like a chain/rope connecting membrane proteins and locking them in place. Spectrin connects to the Band 4.1 protein (also to actin, but less important).
Spectrin also binds to Band 3 protein (via ankyrin, just a protein that connects spectrin to band 3). Band 3 is an anion channel (for HCO3- and Cl-, which is important for red blood cells to function. you want these band 3 proteins to remain evenly spaced out, so they are fixed in the membrane to make sure)
It is important for some membrane proteins not to move. Otherwise the cell will lose its function in some cases(like intestinal epithelial cells), imagine if SGLUT-1 (glucose + galactose transporter) moves from the luminal side(facing lumen) to the basolateral side(the other side)... what do you think happens to its ability to take in glucose from intestinal lumen?
Finally as a bonus, some membrane proteins are also fixed in the plasma membrane in plants (does not move around) by cell wall.
Anyway, if you are bored of someone immediately saying "fluid mosaic model means proteins always move around!" then read my answer