Depends on what is meant by 'sour'. IF taken literally (taste) then NO. If the term is used synonymous with 'decay' then over time....yes.
The 'bitterness' of an acorn is determined by it's tannin content. That is why deer prefer White Oak acorns over certain others...since they have less tannin content the deer find them more palatable.
If anything...being exposed to water would tend to let the tannin leach out of an acorn making it less sour. The Native Americans soaked acorns before grinding them for just this reason.
IF we use the term 'sour' to mean decay/decomposition....then rain combined with the nut sitting on the ground is naturally going to result in Pathogenic Fungi and Bacteria affecting the outer shell and eventually the nut inside. But acorns are by design a durable item. A healthy acorn should remain viable for several months... rain or not.
If you think about it, many places... water (rain) is actually a dispersal agent for acorns. Transporting, relocating them great distances from the tree from which they fell.
If rain were to quickly 'ruin' an acorn then they would seldom have a chance to sprout anywhere except in a close proximity to where they dropped. Squirrels and rain water move a lot of acorns in many environments.
As concerns a 'healthy' acorn.....I doubt much more than 1/2 the mast crop falls to the ground without an acorn Weevil/Grub inside of it. We've all picked up an acorn with an obvious small hole in the shell. That hole is where the grub bored its way out.
These acorns typically look unmolested suggesting that Deer/Hogs/other have passed them over. I think all too often these acorns are thought to have 'soured'.
Not unlike other myths and partial truths, these things get passed on from one generation to the next without much thought given to it.
My Mother-in-Law to this day will argue that the foamy pod found on some Dew-Berry vines is 'Snake Spit' and not simply the protective covering and result of a 'Spittle Bug' doing its work, despite me having shown one to her. So a word to the wise. Some Old Wives Tales and traditions are best left alone, lest you yank some childhood beliefs out from under them.