Looking to upgrade Sora 9-speed to Tiagra 10-speed 52/36t
Shimano's compatibility documentation says there is no compatibility between components of groupsets of different 'speeds'. E.g., there is no compatibility between 3x9 or 2x9 Sora and 3x10 or 2x10 Tiagra. The specific reason a '10 speed' front crank is not compatible with 9 speed group is that chains get progressively narrower as the number of rear speeds goes up and the spacing between sprockets decreases. The chainring spacing on the crank is optimized for the chain width.

That said, the concensus on this site seems to be that you can get away with a 1 speed difference in the crank. 

Rear shifters and derailleur do not need to be changed. You do of course need a double front shifter and derailleur, the triple front shifter and derailleur you have now won't work.