Heating the metal will cause grain growth but it is the large grains that grow by absorbing the smaller grains. The total volume of the grains does not change. It addition if you heat it up it will oxidise and you will lose diameter when the oxide scale is removed. There are other factors to consider as well. Firstly there is the possibility of distortion and secondly the metal will be softer.
You do not say what the application is but a very light knurl will increase the size of the shaft and maybe solve the problem.
Mike