There are a few examples of purchasing:
1. Heading to a grocery store with a list of items that you frequently buy that are priced at similar prices at every store.
2. Going to a store to purchase a specific item
3. Going to a store to find an item
For case #1 it's most likely that you're shopping at a place that is already competing for the lowest price point (places like Walmart and Target) so this application would be useless to you; however these stores already engage in the sort of activity the OP discusses.
For case #2 if it's an item you've purchased previously you probably have a "favourite" store, one that you used previously and had a good experience with so naturally you go to this store. If this is an independent store they might have the item for $5 more than another store of a similar type, but because you've used this store before and because it has always been cheaper on previous occasions you bet that extra $5 against the time investment of finding if it is cheaper elsewhere (which isn't guaranteed) and go with the "safe" option.
If in case #2 you didn't have to consider your previous purchases, if you could just type in the product into your phone and be told (without doubt) that the product is $10 cheaper at a store across the street from your normal store most people would take that; it's $10 that would be saved by walking across the street, it would be stupid not to, right?
So now you have a situation where in case #2 any store that stocks an item (HP Printer, for example) is required to stock it at the lowest price because if they don't and a competitor is within 2 blocks (the "distance of $10") people will just go there instead. So now where is the value to the retailer? They enable themselves to be pitted against their competitors, thus requiring they lower their margins and inevitably lose out to bigger stores that can work with lower margins.
In example #3 the same situation occurs, you enter a store and locate the item that you want, enter it into your phone and get told exactly where it can be acquired and at what price; if the store that has it cheapest is 2 blocks away and the price difference is $15 why not go over to the other store and save the $15? That store just lost a sale because of the system.
Almost everyone that I know (I'm not from a rich background) would absolutely walk over the street to save $15, that's a non-neglible amount of money to quite a large portion of consumers. Sure if every other store has the item cheaper but it's only $0.25 difference people probably aren't going to go to another store, but then it becomes a race to the bottom. Wonderful for the consumers, awful for the retailer.
What incentive is there for a retailer to force themselves to reduce their own margins? Currently purchasing an item in a retail store is a gamble on whether or not you'll pay the lowest price (unless you spend time researching every stores pricing), stores make money because of this. Currently it's a matter of taking a consumer, showing them a product and convincing them to purchase, if this system existed it would have an extra step: convince them to purchase at that particular store.