[Note] The question has been asked at Matheoverflow. But there is no answers.
I believe that any non-trivial idea will sooner or later find application in real life. However "sooner" is better than "later":)
If we look at famous open problems - e.g. Millennium Prize problems - there is no clear practical outcome from their solution (imho). (Well, P vs NP of course will be of greatest importance if one will produce quick algrorithm to solve all NP problems - but it seems it is not plausible).
Question what are open problems which solution will have some "real life" outcome in visible future ?
My answer. It might be not perfect from various sides, but still...
Problem Construction of short and effective error-correcting codes.
Background Recent advances in error-corrrecting codes are LDPC and turbo-codes - invented in 90-ies (more recent polar codes) produced codes which are highly effective for quite a long length of the codewords (thousands bits or may be even dozens thousands). These codes are widely used in practive, e.g. 3G-smart-phones use turbo-codes. (See wikipedia articles on turbo-codes and LDPC)
However such codes are not effective for short sequences of bits (dozens bits). That is why a problem.
Practical importance In cell networks not only data is transmitted, but various auxilliary information - e.g. acknowledge receipts (data has been received or not), various channel quality indicators (measuring how bad are propogation conditions). Amount of such data is not big - dozens of bits - so we need short codes. Up to recent times this was not considered as important issue, because it might be just 1% of data transmission. But with the advance of smartphones and "always on-line" feature this appears to be critical problem. Since "always on-line" means sending very often and very short amounts of data, but for each data transmission one needs to send also auxilliary information - so the amount of auxilliary information becomes comparable with data transmission.
Literature This problem is mentioned in influential article in telecommunication community "Is the PHY layer dead?" http://repositori.upf.edu/handle/10230/13026
("PHY(=physical) layer" - is a part where most mathematical-consuming algorithms were concentrated (e.g. error-correcting codes, statstical algorithms for signal estimation etc.)). So the title has a flavor "math is not useful for wireless telecom anymore?" (However now RM (Resource managament) layer is consuming math (from game theory to optimization) so we may not worry :).