Christine , for the regular visitors , you cannot go to Lebanon if there's a proof on you passport that you've been in Israel. The immigration agent will refuse you enter the country . And I find this to be extremely stupid - there are many more ways for spies to enter the country ( because that's the reason being given by Lebanese authorities). Unfortunately The only way for a tourist to visit both countries is to ask the Israeli authorities - if they accept - not to stamp the passport , and then to leave Israel via Jordan. Or the tourist might visit Lebanon at first and then enter Israel through Jordan and beg the israeli authorities to permit entering the Israeli lands , they might keep the visitor waiting for hours at the airports to investigate.
On the other hand , VIPs can easily go from Israel to Lebanon ( via Jordan ) without any problem. Armin Van Buuren was doing a concert in Tel Aviv before coming to his Beirut's concert. And I think it works the other way back.