In the response to List of Techelet Wearers it mentions a few Gedolim that wear Tzitzis of Techeiles on Shabbos only. Why would there be a difference between Shabbos and the rest of the week?
I can come up with a few reasons of why to only wear it on Shabbat.
You wear the best clothing on Shabbat. Techelet is Expensive, and to wear it every day might be too expensive to clean properly or to buy extra garments. However, on Shabbat you need to wear your best, so you buy a set of techelet for Shabbat.
They only bought one set of Techelet strings, and wanted it to be attached to their favorite talit/jacket.
I know a rabbi who only wears on Shabbat because he thinks it's a reasonable chumra but he doesn't want his kids to grow up thinking that we have a mesora for techelet so he only wears on his Shabbat/Yom Tov Tallit Gadol. As for why then, he said that Shabbat is a traditional time to be more makpid on certain chumrot (for instance, putting salt on the bread that we eat).
There is a famous "brisker chumra" to be worried about the position of the Ba'al HaMa'or who said that tzitzit are only kosher if they have the blue and the white (just white is just not good enough) and refrain from wearing their tzitzit outside without an eruv as according to him, the tzitzit are invalid and you would thereby in fact be carrying the strings for no reason! Maybe some people wear techelet on shabbat so they don't have to worry for this opinion.
Well, for one of the Rabbanim- Rabbi Meir Mazuz Shelit"a it is because he wants to show that it is not a Hiyuv.