Has anyone burned hydrogen in a gas turbine?
Yes, indeed the plan to build a hydrogen-powered jet laid the groundwork for using hydrogen in the Centaur rockets and the upper stages of the Saturn 5. While hydrogen was used only experimentally in test rigs, the Soviet Union built a derivative of a regular airliner, the Tupolev 155, for testing hydrogen and natural gas in flight.

EDIT: NACA also did in-flight testing. Please see @jayhendren's answer for details.

So yes, the Tu-155 was indeed flown on several occasions with one hydrogen-powered turbofan. While the left and center engine remained NK-8s, the right engine was replaced by a NK-88 which was adapted for LNG and hydrogen. Other projects like one for a hydrogen-powered supersonic airliner sadly ended with the Soviet Union itself.



Tu-155 Cutaway view (picture source)

Hydrogen is also the prime propellant in hypersonic ramjets, but those are not turbines.

Hydrogen has a wide mixing ratio with air where it will burn. Also, being gaseous, it mixes much more quickly with air, so the combustion chamber can be small. When a converted J-57 was experimentally run on hydrogen in 1957, 


  The test engineers were agreeably surprised by the ease of engine operation. They ran it at full power and throttled back so far that the air fan was revolving so slowly the individual blades could be counted. Under this latter condition, the throttle could be opened and the engine would quickly and smoothly accelerate to full power. They found that the temperature distribution was good and there were no major problems.


But Niels is correct - its low density makes hydrogen problematic. As Alexis W. Lemmon, Jr., reported in May 1945 in his report on possible jet fuels (from history.nasa.gov):


  "Although the liquid hydrogen-liquid oxygen system has by far the highest specific impulse performance of any system considered in this report, the low average density of the fuel components almost completely eliminates this system from all but very minor applications."