In larger cartridges with high neck tension, the primer can become a projectile. Primers have enough power to launch themselves with enough velocity to penetrate skin, and do severe damage to eyes. They should be considered to be as dangerous as a pellet rifle being fired.
In reality, the primer is rarely ejected from the cartridges. It's that low chance, that makes you weigh the odds for yourself.
As far as the case and bullet... Bullets usually don't go far. For most cartridges, the bullet weighs more than the casing. The casing will move significantly farther. ...But even 3-6" of casing movement can be significant, when the bullet just pushes forward 1/2", and rolls over.
Case ruptures are the other issue. Most cartridges 'blow' by having the casing pushed off of the mostly-stationary bullet. But, when they decide to rupture, you get shrapnel. Those sharp, unpredictable pieces of metal can be nasty (as well as hot embers, if you toss ammo in a fire... or build a fire on some one's buried ammo cache
{don't ask}).
...Which brings us to firefighters.
They wear protective clothing and equipment that will stop all of these hazards. One of the towns I used to live in actually went so far in their "Home Hazards and Firefighter Safety" pamphlet*, as to tell people not to worry about storing ammunition in their homes. They elaborated that the firefighting gear offered more than enough protection, and it was a non-issue.
But, as we all know... They spent a significant amount of time warning about the storage methods, locations, and quantities of: Paints. Oils. Gasoline. Propane. Cleaners.
Any aerosol cans. Alcohol. ...and the list goes on. They even devoted a full page to proper selection of flame-resistant bedding materials; but ammunition was 'don't worry about it'.
(*I was trying to find any local restrictions on smokeless powder. They directed me to this pamphlet. Later that year, the actually had a demonstration where they dumped small arms ammo into a fire, then walked up and put it out.
)