In the US, the term "dress shirt" is a category and not a specific item of clothing
So it is in the UK: but here it's a quite different category. It covers any shirt designed specifically for evening dress, either "black tie" (black bow tie and dinner jacket) in which case it may have ruffles down the front as Chez Miriam says, or "white tie" (white bow tie, starched white waistcoat and tailcoat), in which case it is fastened down the front with removable studs - necessary because the shirt has to be boil-washed and the front starched rigid, a process which buttons would not survive - and has a separate starched collar fixed to it with more studs.
I think we normally refer to that as a tuxedo shirt. Could be other names, but that is the only one I'm aware of.