Taxis in Beirut

Taxis aren't metered. The going price for door-to-door taxi trips in town is 8,000 to 10,000 LBP. If you look like a tourist they may well ask for more, eg 10 USD (15,000 LBP). Fix the price before getting in. If you don't like the price just walk away - there are so many other taxis that will come honking for your custom.
The buses cost just 1,000 LBP in town. The shared service-taxi trips are 2,000.

For the airport it is likely to be around 20 to 25 USD.

gingerbeirut.com

For a short rides in town I haven't fixed the rate before usually. Just giving the normal price after the trip like locals are doing. But I haven't been to Beirut for two and half years now, so it's good to get an idea what the "normal" prices are currently :)

There is no change to get the taxi to airport for 15 000 or 20 000 LBP anymore like two years ago? So the minimum price whould be 30 000 LBP?

As far as I can work out, if you look like a tourist, or you hail a taxi rather than waiting for them to come to you, best fix the price before, unless its a shared trip, in which case it's quite obvious you won't be paying a lot. But if you can get by in Arabic, and you know where you're going then there's no need.
From what I hear most people end up paying about 25,000 LBP to the airport and 30,000 from the airport. But if you call and book in Arabic, you can get as low as 15,000 LBP going to the airport. If you can get a couple of taxis to compete then you can also bring the price down but you may not be able to do that if you want very early morning.
gingerbeirut.com

Thanks a lot. I'm not looking like Lebanese but I can act with taxi drivers like a local resident when needed and can use some Arabic also. So far I haven't had any trouble when taking a taxi. You just have to stay away from hotels entrances and so on. Maybe only at the night when driving from pub to club :)
Waiting on for trip to Beirut. Two years without a visit have been too long

If you hail any taxi passing by (with a red licence plate - usually old and battered Mercedeses), you tell the driver "servees" which is 2000 lbp. Usually Hamra to downtown is 2 x or 3 x servees.