I've never seen sticky ash. I've seen completely beige and brown nicotine on walls, but never sticky.
This is more likely due to cooking. If the cooking hood/extractor doesn't work right or they just haven't cleaned in a while, the fumes from cooking will cover everything in a thick layer of sticky fat, which is difficult to get rid of with normal cleaning products.
Cigarette ash absolutely gets sticky from tar. Source: worked in a casino for a decade, have seen dust built up in PC fans and any other airflow device or crevice many times that looks somewhat like fuzzy sludge. Fucking disgusting.
And yes all the beige curtains, walls, etc were white when we first put them in, matter of weeks for a brand new white curtain wrap(used to hide the poles for chandeliers that have camera arrays in them) to turn tan/beige
It's not the ash, it's the tar. Smoke lingers in the air, cools, and condenses on surfaces including walls. It creates a fine film that sticks to everything. Sometimes it seems like it's even embedded into the paint but I'm no materials engineer.
Source: I grew up around chain smokers and my childhood best friend has 2 parents that constantly smoke indoors.