How much would it cost to go from Lijiang to the tiger leaping gorge by taxi?

Well, I'm afraid that there is no taxi. You are advised to hire a private car to TLG directly. The fare would be around CNY 300 for the single trip.

Ride a bus for 40 Kuai/RMB to Tiger Leaping Gorge. Either flag down a bus or catch a ride on a mianbaoche (small van) back for a similar price range. It took about 30 minutes for our ride to start back. This is the best way if you plan to hike.
If you want to ride through the gorge on the road, then maybe look into a tour. You could stay with the same bus the whole way. If you are in a hurry, the fastest way would be to hire a mianbaoche for the day. The ride isn't as smooth as a bus, but they do move quickly. The going rate to rent one out for the day based on other places I've gone in the area seems to be around 500 Kuai. You get the whole van and can divide out the costs if you can get others to share it with you. They usually fit 7 passengers.