According to this video on jet engine blades, titanium is never used for the turbine, because it "melts and burns at the temperature of the flame". Only the blades of the compressor are made of this material.
However, the melting temperature of titanium is higher than that of any steel alloys I could find, and titanium produces a protective layer of oxides on the surface. What other properties of titanium make using it impossible?
Titanium is unsuitable because it will react with oxygen and carbon at high temperature, well below its melting point, making it very hard and brittle. Welding titanium is very complicated because it needs to be shielded extremely well from any oxygen when hot. Ti$_3$O will form above 500°C, and Ti$_2$O above 600°C.
Initially turbine blades were made of steel alloys, but they have been displaced by nickel alloys.
Also, they operate in an environment which requires constant cooling, so they can be 200 - 300°C cooler than the turbine entry temperature of the gas coming from the combustor(s). Modern turbine blades are hollow and have a perforation at their leading edge. Pressurized, relatively cool air is forced through the blades and the perforation and flows around the blade's surface, creating a cool sheet of air to shield the blade from the hot gas. Also, before entering the turbine the gas is accelerated, which already lowers its temperature. See the plot below of parameters inside an older engine taken from this source.
Shortly past the fuel injectors the maximum gas temperature of approx. 1800°C is reached, which drops to 1100°C at the entry to the first turbine stage. Note that this temperature has been raised to 1500°C in modern military engines! At the same time, the highest temperature is connected with the lowest speed (30 m/s), and the flow accelerates to 200 m/s directly before entering the first turbine stage.
Titanium in contact with oxygen would lose a lot of its strength at these temperatures, even though its melting point is at 1650°C.
In addition to cooling and nickel alloys, two other technologies are used: monocrystalline casting and thermal barrier coatings.
Thermal barrier coatings, in concert with cooling, allow operation at close to the melting point of the base material (as referenced above). TBCs usually consist of yttria-stabilized zirconia, which has very low thermal conductivity and a coefficient of thermal expansion close to that of nickel alloys. This makes it incompatible with titanium, which has a lower CTE; The different growth rates induce stress into the coating, eventually cracking it.
In addition, YSZ is oxygen-permeable at high temperatures. This can be reduced (temporarily) by an underlayer, but eventually oxygen penetrates to the substrate. Nickel alloys are substantially more oxidation resistant at temperature than titanium, as previously mentioned, so they also are more compatible with TBCs in this way.
The other technology used in hot section rotating components is monocrystalline casting. Simply put, larger crystals resist creep because they are less likely to combine with other crystals, and you don't get larger than a single crystal. To my knowledge, there is nothing about titanium casting alloys that makes them incompatible with single crystal casting.
