You can use chisels and other tools that were not specifically made for turning but you shouldn't expect their edges to hold up for very long. Even with commercial turning tools made from steel frequent sharpening is the norm, so with anything less robust than this you're going to be sharpening a lot so be prepared ahead of time.
As much as the hardness of the wood being turned the speed of rotation and the size of the piece are factors — remember that for any given RPM the edge of a larger piece of wood is moving faster than a smaller piece of wood, so it becomes more abrasive and there is greater chance it can fracture or crack the cutting edge of a tool.
Some related info in a couple of previous Q&As:
http://woodworking.stackexchange.com/questions/4324/can-i-use-sculptor-chisels-with-a-lathe/4330?s=3|0.3842#4330https://woodworking.stackexchange.com/questions/4324/can-i-use-sculptor-chisels-with-a-lathe/4330?s=3|0.3842#4330
http://woodworking.stackexchange.com/questions/2617/what-should-one-consider-when-purchasing-an-intro-turning-chisel-set/2619?s=2|1.9627#2619https://woodworking.stackexchange.com/questions/2617/what-should-one-consider-when-purchasing-an-intro-turning-chisel-set/2619?s=2|1.9627#2619
http://woodworking.stackexchange.com/questions/3374/how-can-i-tell-if-wood-turning-lathe-chisels-are-sharp/3377?s=1|2.0486#3377https://woodworking.stackexchange.com/questions/3374/how-can-i-tell-if-wood-turning-lathe-chisels-are-sharp/3377?s=1|2.0486#3377
