mamashtakah is absolutely correct but if we were to apply his standard to this place most posts are in violation.
the thing with techeiles is as follows: there are no halachic tools available to poskim to assist them in determining whether one should go out and wear it. the reasons to wear tekheleis are purely scientific.
that being said, it is clear among all poskim that there is no halachic violation in wearing blue non-tekheleis.
The kabbalists, however, caution against wearing blue and believe that in galus we must wear white for kabalistic reasons (something to do with blue bringing middas din on us which we cannot handle in galus).
the kabbalistic interpretation is troubling in some respects as it appears to use kabbalistic reasons to nullify a mitzvah de-orayssa
the kabbalists can maybe be understood to mean that the tekheles is surely not genuine because hashem has a reason to hide it from us.
in any event, since all of us wear blue shirts, suits, socks, etc., we do not live by the words of the kabbalists anyway.
maskunu: there is no way halacha can obligate us to wear it but it does not hurt to do it.