In general, if you window the game, move it to the other monitor, and then full screen it, it should continue to open on the correct monitor. This is how I have solved this problem in the past, and it has worked for every game I have tried it on.
Alt+Return seems to be the most reliable. I have also used Alt+Tab, Ctrl+Esc, and Win+Down Arrow to get a game ...
First try -safe and -autoconfig (separately)
Right click Dota 2
Select 'Properties' from the menu
Click the 'Set launch options...' button
Remove any launch options currently shown
Type -autoconfig or -safe in the box and select OK
If those don't work then try
use your monitor's current settings
changing your rendering API (note ...