iOS / C: Algorithm to detect phonemes
If you are looking for a phone-level open source recogniser, then I would recommend HTK. Very good documentation is available with this tool in the form of the HTK Book. It also contains an entire chapter dedicated to building a phone level real-time speech recogniser. From your problem statement above, it seems to me like you might be able to re-work that example into your own solution. Possible pitfalls:
Since you want to do a phone level recogniser, the data needed to train the phone models would be very high. Also, your training database should be balanced in terms of distribution of the phones.
Building a speaker-independent system would require data from more than one speaker. And lots of that too.
Since this is open-source, you should also check into the licensing info for any additional details about shipping the code. A good alternative would be to use the on-phone recorder and then have the recorded waveform sent over a data channel to a server for the recognition, pretty much something like what google does.
I have a little bit of experience with this type of signal processing, and I would say that this is probably not the type of finite question that can be answered definitively.
One thing worth noting is that although you may restrict the phonemes you are interested in, the possibility space remains the same (i.e. infinite-ish). User training might help the algorithms along a bit, but useful training takes quite a bit of time and it seems you are averse to too much of that.
Using Sphinx is probably a great start on this problem. I haven't gotten very far in the library myself, but my guess is that you'll be working with its source code yourself to get exactly what you want. (Hooray for open source!)
...using a sledgehammer to crack a nut.
I wouldn't label your problem a nut, I'd say it's more like a beast. It may be a different beast than natural language speech recognition, but it is still a beast.
All the best with your problem solving.