The gemara in Kesubos 28b seems to say no. The gemara there is discussing testimony which allows a person to eat Teruma. The gemara brings a story with R' Elazar where he states that he gave testimony that allowed an עבד כנעני to be mistaken for a כהן. The gemara asks from the famous story in Chullin (7a-b) with R' Pinchas ben Yair that if Hashem doesn't let the animal of a צדיק cause a mishap קל וחומר a צדיק himself won't cause a mishap. Therefore the gemara corrects the story that the beis din was going to pasken that he was a כהן but didn't actually do so. We see from here that this principle applies in general to צדיקים.
However, Tosafos in a number of places in shas (Shabbos 12b, Gittin 7a and more) disagrees. Tosafos is bothered by a number of cases where we see that צדיקים did cause mishaps by their actions (for example in Shabbos 12b, R' Yishmael learned by the light of a candle and tilted the candle and violated an issur d'oraysa). Therefore R' Tam posits that this principle only applies to matters of food. Hashem will not cause the a צדיק to eat or violate an issur relating to food. However, this principle does not apply to other areas. R' Tam is therefore forced to change the girsa of the gemara in Kesubos and take out the line which refers to the principle. The Netziv suggests that the reason that this principle only applies to food is that non-kosher food is metamtem es halev.
The Ritva in Shabbos disagrees with R' Tam and answers the case in Shabbos by saying that R' Yishmael was punished for vioating the gezera d'rabbanan and the Ritva rejects R' Tam and claims that this principle applies in all cases.
In summary, there is a machlokes rishonim whether the principle that the actions of a צדיק do not cause a mishap applies to all cases and all kinds of mishaps or just to food related aveiras.