Colour film is perishable. It has a shelf-life and it will go bad, though not in a spoiled-vegetable sort of way. This is why reputable film manufacturers will encode film products, including single-use cameras, with a "best before" date. Like produce, how film is stored has a lot to do with how well it performs after its expiration date.
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Colour film is made of three or more colour dye layers. Each dye has its own properties and much research went into each film brand to balance these factors to give realistic results. As film ages, each dye layer ages at a different rate, and the balance is shifted. How it shifts changes from brand to brand, but generally prints made from expired film stock will exhibit a magenta shift, showing a pink tinge to neutral subjects like clouds.
Exposure and Base Fog
Conventional film uses silver halide technology and this, too, is perishable. The light-sensitive silver layers for each colour will lose sensitivity over time, although not in a predictable way. In a disposable camera this problem is compounded by the inability to change exposure. Also over time, silver emulsions gain their own density, what is called "base fog," and essentially makes the film less transparent, so prints have dull looking highlights.
How your disposable camera is stored has much to do with the results you achieve. Heat is a big enemy of colour film. Sticking a disposable camera in the glove box of your car in hot summer weather can cause film to degrade even before its expiry date. On the other hand, photographers have for years been buying up near-expired film at low prices and freezing it. This preserves the film, but it introduces a problem of condensation if not thawed carefully.
Some disposable cameras have built-in flashes, powered by a single AA battery. In a new camera, this battery holds sufficient charge to expose the film, but like all batteries, it will lose some power as it ages. A camera past expiry may not be able to charge the flash capacitor. However, because the cameras use a capacitor, a weak battery can still build sufficient charge, so if the flash indicator comes on, you need only worry about the film's sensitivity, not the power of the flash.