In this teshuva written to the Chafetz Chaim, the Netziv takes issue with a psak written in the Mishna Berura (which was written by the Chafetz Chaim) where he writes that if one has a tallit which is pasul for some reason or another then he may not go outside on Shabbat with that tallit (in a place with no eiruv), and if he does he will be in violation of the prohibition of carrying on Shabbat (presumably he is considered to be carrying the strings).
The Netziv notes that the view of the Mishna Berura is against the overwhelming majority of Rishonim and Acharonim. According to Rashi, if one had techeilet that was not fit for the mitzva then he would violate the prohibition of carrying, but if one only has white strings they are not considered to be that important and thus there is no prohibition (remember, the Netziv lived before the revival of techeilet). According to Tosafot, there would only be a violation if one had a garment with three tzitzit made already and the fourth was missing.
The Netziv does point out that Rav Akiva Eiger leaves open the possibility that one could violate Shabbat with any form of psul in his tzitzit, but the Netziv counters that other than that possibility, there does not seem to be any major support for the position of the Mishna Berura. There also seems to be a view in the Rishonim that tzitzit should not be worn at all on Shabbat for this reason, but the Rosh rejects this as an unnecessary stringency.