Can I run 2x10 XT Shifters on a flatbar road bike with 105 10 speed STI components?
You have a few options for replacing your STI shifters for use with flat bars (but not the XT shifters for the reasons in the comments):

 You can use Shimano SL-R780 Flat Bar Shifter 2x10SP shifters with 105 components on flat bars. 

http://www.shimano.com/publish/content/global_cycle/en/au/index/products/road/flat_handlebar.html

You could get some 10 speed bar end or downtube shifters and use those with some paul's components thumbies. 

http://www.paulcomp.com/thumbies.html

 You could get 10 speed bar end or downtube shifters and mount them on the downtube or stem with some problem solvers downtube shifter mounts.  

http://problemsolversbike.com/products/downtube_shifter_mount1