Most of us may have seen similar imagery:
How could a player character do as such, with a shield?
Anything RAW, and without resorting to spellcasting (if at all possible), would be nice.
A creature fighting a dragon can take the ready action specifying the action as I gain total cover from my tower shield and the condition as When the dragon is about to use his breath weapon. When the condition's met and the dragon uses its breath weapon anyway, most dragons' breath weapons no longer have line of effect to the warrior because the warrior has sought refuge behind a barrier.
Although realistically (or as realistically as this situation gets, anyway) the tower shield (hardness 5, 20 hp) should at least be damaged by many a dragon's breath weapon, because most dragons' breath weapons don't say they damage barriers (cf. lightning bolt), the tower shield remains unscathed afterward.
So while tower shields have a poor reputation, this is one of the handful of ways they don't suck.
Yes, if it's designed in some way to resist such an attack. If it's a line effect breath weapon, like with a blue dragon's lightning, or something to that effect, then there's no reason a large shield wouldn't at least help against said damage. But a mundane shield would take damage and not necessarily negate anything that goes outside the shield's dimensions - ie the guy next to you would still be in the area of effect, if it's a cone and not a line. If your DM's version of the dragon only does a line breath, then sure, why not. But I'd say anyone right next to you would, all things being equal, take half the effect's damage in splash, because a shield used in this way does, in art, make the breath weapon go off to the sides instead. Deflection, in other words. Like how a surface would deflect a projectile. If it makes sense for a shield to deflect an arrow, then it would make sense - as much as a mundane shield protecting vs a line breath would - to parallel this if done correctly (hence readied action and breath condition).