I made that mistake.
I buy Sunbeam or Panasonic heavy duty and never had a problem.
I bought Panasonic batteries at the Dollarama (a Dollar Store in Canada), for my computer mouse. I thought my computer mouse was deader than dead, then my Dad puts a Duracell battery in and it is fine now.
they won't last near as long as good batteries, so in the end you probably will spend more on batteries or the same at least as you woul dif you went more premium
I wouldn't buy them there. I buy Duracell only. I find they last the longest of any other battery I've tried.
I'd say no, but it depends. If you read the label closely, sometimes the store brand batteries are actually made by a reputable company and re-badged. Other times they're just cheap made in China junk. Also, stay away from "heavy duty" batteries; these are carbon zincs which don't last nearly as long as Alkalines. They may cost a few bucks less but you'll spend a lot more money constantly replacing batteries. They've been obsolete since the '60s, and Alkalines are superior with no downsides.
The MAIN reason to avoid cheap batteries though, is because they have a tendency to leak and ruin your device. Even good batteries can leak sometimes, so it's always a good idea to check the batteries in your devices every once in a while.
As for rechargeables, I've never actually bought rechargeable batteries at the dollar store, but I'd avoid them like the plague unless I read the label and made sure they were made by a reputable company. Cheap rechargeables seem like they would be a lot more dangerous because something could happen during the charge cycle.
I usually get Duracell or Energizer batteries.