No, don’t compost old wallpaper.
While paper is usually compostable, it’s better not to compost old wallpaper because it is rarely just paper.
Most wallpaper these days includes some sort of vinyl coating (especially wipeable wallpaper or embossed/textured wallpaper) or in the case of many feature wallpaper designs, include foil or vinyl inlays. These will not breakdown in a compost heap.
Plainer wallpapers – such as lining paper or old school woodchip – are typically painted — the paint might also be vinyl based but even if it isn’t, you don’t want the chemical residues from paint in your compost.
Finally, while wallpaper paste are often modified starch or cellulose based, they often include synthetic additives which you might not want in your compost heap. (There are also specialist vinyl based adhesives for use with vinyl wallpapers – as you might guess, they’re a definite no-no!)