Absolutely it can, and in a variety of ways.
Modern cars, with all their electronics rely on a fairly steady electrical current with few spikes and lows, a battery in a very real sense acts as a current modulator/stabilizer and if weakened can cause an erratic current which the electronics don't like.
Also, a weakened battery will reduce your mpg as the alternator is constantly having to send it a charge, draining crank hp - as much as 6-10hp.
Of course, losing 6-10hp to the alternator will affect the car's performance as well.
A battery is good for 3-5 years under OPTIMAL conditions. After that, it's performance and effect on the rest of the car is questionable. Under less than optimal conditions (severe cold, long storage periods, occaisional discharges, rough handling or severe shocks) expect the battery to last much less long.
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