It is difficult to answer this question because the word “tainted” is hard to employ in a precise manner. Let us divide the discussion into two parts, halacha and philosophy.
Halachically, there is a concept that a mitzva can be nullified if it is linked directly to the violation of a prohibition. One example: if one steals an etrog and then uses it on Sukkot as the Torah commands, he has not fulfilled his mitzva. However, this rule does not apply here because the linkage between the sin and the mitzva is not direct enough. (If you like, we can present a Talmudic discourse to explain the distinction, but this is the bottom line). Thus, the tzedaka is not halachically tainted.
The mitzva of tzedaka, like, perhaps, all other mitzvot, is not just a perfunctory act where the only factor is how much a person gives. Notably, Maimonides goes into much detail about eight different levels of giving tzedaka (he concentrates on levels of avoiding the pauper’s embarrassment). Part of the philosophy behind giving away a portion of one’s earnings is that he thereby demonstrates that he realizes that HaShem ordains all of his blessings (including his earnings).[1] If one earns a portion of the money that he donates in a manner in which he transgresses HaShem’s commandments (e.g., by violating Shabbat), that element of giving tzedaka is incomplete. Sometimes, one does realize that it is HaShem who controls a person’s successes and failures and is remorseful that he has not succeeded in fulfilling His will. If he thus sees his tzedaka as a way of saying “thank you” and “sorry,” respectively, then we have a mixed bag. On one hand, that is wonderful, as this mitzva, with the proper intention, can bring him closer to his Maker and Provider. On the other hand, he should not see it as a reasonable trade-off (“I’ll continue to work on Shabbat, but it is okay because part of my money goes to tzedaka”). If he does, then the tzedaka could be considered an “accomplice” to the continued violation.
Leaders of Torah institutions have been torn over similar issues. On one hand, one cannot “dismiss” Jews because of their shortcomings in the performance of some mitzvot, even crucial ones, especially in light of the very weak religious climate by which most Jews are surrounded. Shouldn’t all Jews be wholeheartedly encouraged to perform good deeds (tzedaka is a great one) and be praised for doing so? Additionally, the connection to HaShem and the spiritual elevation that so often accompany sincere, heartfelt donations to worthy institutions can often bring the person closer to Him. A small minority take the opposite approach, claiming that the responsibility to rebuke those who sin precludes our embracing those who are still in the midst of sinning. Otherwise, they believe, we encourage wrongdoers to continue without correcting their ways.
As spiritual disciples of the great Rav Kook (our mentor, Rav Shaul Yisraeli, was a close student), we believe in embracing all Jews, regardless of their level of religious observance. Certainly, we believe in encouraging the fulfillment of any mitzva and will not say to someone, “All or nothing.” Nevertheless, one should not mislead a Jew by telling him or implying that “as long as you give money to institutions or people we deem worthy, you are fine in HaShem’s book.” We don’t have access to HaShem’s book of the righteous and His book of the sinners. If we did, I am sure we would find many surprises. We do have access to HaShem’s book of instructions for a proper Jewish life – that is the Torah. It is the responsibility of all of us to both fulfill those instructions ourselves and to help others do so. It is not easy to do that perfectly, but we need to try our best.