I'm super grateful for these insightful response.@Lusco,
For most machines with diesel engine starting means driving the shaft through a torque converter the answer really depends on the cooling water system for the diesel engine. Some machines have a small cooling water system tank which is good enough for one or two starts and maybe a half-hour or one hour of running the diesel at near rated output. Other machines have a larger diesel engine cooling water system tank and can run for longer periods of cranking.
Some machines run the speed up to a higher speed for a short time during purging then reduce the speed in preparation for firing--and every time one initiates a START, even in CRANK mode, the machine goes through the same sequence on every START. Most Frame 5 GE-design heavy duty gas turbines use the starting means to run the machine up to purging speed and once the purge time is complete the machine continues into firing mode at the same speed when STARTed in AUTO mode. These machines, when STARTed in CRANK mode, will run up to a speed that is at a high load for the starting diesel--which require a higher cooling water system flow, and tank. Some machines actually get cooling water from a cooling water system used for other purposes on the site and therefore have an unlimited supply of cooling water and can run for very long times when in CRANK mode.
So, without knowing a lot more about your machine than you provided, we can't give you a specific time you can run while CRANKing. It's really about how "robust" the diesel engine cooling water system is, and, of course, how much diesel fuel you have.
As far as hurting the turbine-generator, it's not going to happen. The torque converter can get quite hot if there isn't cooling water for the L.O. system running/available, and the L.O. will also get warmer and warmer just due to the heat of the Aux. & Main L.O. pumps running and the flow of oil through the machine's bearings--so having the L.O. cooling water system available and running is also important and can be a limiting factor. If the L.O. is already warm because the machine has just been shut down or tripped then if the L.O. is allowed to get too hot the machine will be tripped and unable to be started until the L.O. temperature cools sufficiently.
So, that's about all the conditions that one has to be concerned about which will affect how long a machine with a diesel engine-driven starting means can be used to CRANK the machine--basically the diesel engine cooling water system, and the machine L.O. temperature and L.O. cooling water availability. Other than the fuel for the diesel engine, it's hard to hurt the machine by CRANKing it for a long time. (Of course damage can be done by forcing logic and opening the turbine compartment doors if the machine is hot after shutdown or a trip (the exterior turbine casing can cool faster than the axial compressor rotor and cause a rub/rubs (or turning on the turbine compartment vent fan while the machine is cranking when it's hot--neither are recommended if the machine is hot while CRANKing!).)
That's really all that can be said without knowing a LOT more about the machine. While it's difficult to believe, EVERY Frame 5 GE-design heavy duty gas turbine IS NOT the same as EVERY OTHER GE-design Frame 5 heavy duty gas turbine--yes, the turbines all suck, squeeze, burn and blow. BUT the auxiliaries (starting means; cooling water systems, L.O. systems; etc.) can be very, Very, VERY different and affect operations significantly.
Best of luck!