As we have seen, Kabbalistic sources teach that a spirit of impurity descends upon a person's hands overnight, and it is removed only by washing his hands three times in alternating fashion. If a person does not wash his hands in this manner, the impure forces remain on his hands.
This gives rise to the question of whether a person must repeat Netilat Yadayim after shaking the hand of a Jew who most likely did not wash Netilat Yadayim that day. If one knows that the fellow whose hand he shook is not halachically observant, or is not knowledgeable about requirements such as the morning Netilat Yadayim, must he wash his hands again, since the impurity transferred from the other person's hand onto his? (This question applies only in the case of a Jew who most likely did not wash Netilat Yadayim; the spirit of impurity does not affect the hands of gentiles, and so there is clearly no concern when one shakes a gentile's hand.)
Hacham Bension Abba Shaul (Jerusalem, 1924-1998), in Or Le'sion, rules that one does not need to repeat Netilat Yadayim in this case. The only exception, he writes, is if one's hands were still wet from the morning Netilat Yadayim when he shook the other person's hands. According to Hacham Bension, the Halacha in this case resembles the case of one who washed Netilat Yadayim for bread and then touched the hand of somebody who had not washed. In that case, one needs to repeat Netilat Yadayim only if his hands were still wet from the initial washing when they came in contact with the other person's hand. Hacham Bension maintained that after the morning Netilat Yadayim, too, one who dried his hands does not need to repeat the washing if he touched the hand of somebody who had not washed Netilat Yadayim. This is the ruling also of the Kapeh Aharon (Rav Aharon Azriel, 1819–1879).
Hacham Ovadia Yosef, however, noted that Rav Haim Palachi (Turkey, 1788-1868) requires repeating the washing in this case, and that this appears to be the view also of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), since the impurity can be transferred by the hand shake. Therefore, Hacham Ovadia ruled that one should try to be stringent and repeat the washing after shaking the hand of a Jew who did not wash Netilat Yadayim in the morning.
This applies only if one touched the other person's fingers, which is the place to where the forces of impurity cling. If he touched only the palm, then he certainly does not need to repeat Netilat Yadayim.
Summary: Different opinions exist as to whether one must repeat the morning Netilat Yadayim if he shakes the hand of a Jew who most likely did not properly wash Netilat Yadayim in the morning. One should consult his Rabbi for practical guidance.