Answer: The question of whether it is okay for a non related man and woman to shake hands largely depends on whether this contact is classified as a דרך תאוה וחיבה, a ‘way that is based on attraction’.
Lenient authorities have quoted the Shach (YD 195:20), who, quoting Rambam, writes that it is (only) forbidden for a man and woman to touch in any manner that causes attraction (chiba). Where touching yields no pleasure, it would be permissible (YD 157:10). Thus, R’ Moshe Feinstein paskens (Igros Moshe EH 2:14) that one can sit on a tram next to a woman even if they are accidentally touching. R’ Chaim Berlin (Nishmas Chaim, EH) writes that today, shaking hands is a regular way of greeting people. One has to bear in mind what reputation such a refusal will leave Religious Jews with. Rather shake hands than give others the impression that we’re crazy and are lacking derech eretz. R' Chaim Pinchos Scheinberg also allowed it in extenuating circumstances (Halichos Yisroel, p282).
R’ Moshe Feinstein maintains, however (Igros Moshe EH 1:56), that it is difficult to claim that shaking hands is not considered an expression of affection. In another responsum (OC 1:113) he wrote that doing so is forbidden. Likewise, the Steipler (Kreina Deigresa 162) defines hand shaking as touching with chiba, and therefore a most severe prohibition.
In conclusion, even the lenient authorities only allow shaking hands in extenuating circumstances. As leading poskim have forbidden it, one must try their utmost to avoid it.