Barring such out of normal experience wierdness as catastrophically bad ammo or extremely worn/fragile weapons, all that bad ammo is likely to do is foul your gun more. In other words, the lower quality powder and lower quality bullets will leave more carbon and lead deposits causing you to work harder when cleaning.
Cheap ammo may well affect reliability though, and reliability failures may cause mnor damage. If you are firing an auto-loader, a cheap casing may cause a stovepipe (ie when an ejected round does not clear the ejection port and is trapped by the slide/bolt returning to battery), and stovepipes can cause minor damage to your firearm's finish.
The thing to remember is that you firearm is (hopefully) very well built, and designed to work under adverse conditions (all guns, excepting competition guns are designed, in one way or another, to function in adversity), and to accept ammo loaded to a broad range of tolerances. Don't use truly crap ammo and be careful if you have a stoppage/failure.
Last word, some cheap ammos use corrosive primers. It's not that big a deal so long as you religiously clean the bore after every trip to the range, but it is something to pay attention to as corrosives will wear/corrode your bore if you do not properly clean them out after a shoot.
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