Damascus to Tyre

It seems that if you go to Tyre on the coastal road there is no need for a permit. I travelled it last week with some Lebanese and there was no check on papers. However when I crossed the Litani further inland (heading towards Marjayoun) earlier in the year, our nationality papers were checked and signs said foreigners could not pass without a permit. So if you do choose to cross the Litani on the coastal road you should be just fine. A diagonal route from Damascus to Tyre does look shorter, but I don't know those roads. Most buses and shared taxis go from Damascus to Chtaura, then some split ways while most continue to Beirut. Another possibility would be renting a car while in Lebanon. That would certainly be the quickest, if possible for you.
Damascus-Beirut is usually three hours, but its the border crossing that takes about an hour which makes it long rather than traffic.
Not sure I've helped much - see what others have to say :-)
gingerbeirut

You should be fine indeed if you take the route Chtaura - Beirut - Saida - Tyre.
No problem with checkpoints, only as said at the Litanie.
The only thing is that roads are badly lit, so you should drive with extreme care.