I have been running DoTA conveniently under Linux using Wine, since more than an year when I actually started playing DoTA. Everything works perfect without much effort. In this post, I explain the basic steps to get full DoTA Experience.
Very little configuration required. Might run straightaway. But there might be problems which will prevent you from switching between windows, minimizing, lose window focus, taskbar showing on top, etc. The best way to solve these is to configure wine to emulate a virtual desktop, and launch Frozen Thrones in a separate workspace. That way, you can switch between workspace conveniently using keyboard shortcuts (It is more convenient than minimizing under Windows). Execute winecfg and select Emulate Virtual Desktop under the Graphics tab. Make sure to fill up your native resolution there (or the resolution at which you will run the game). Run the game through
nimesh Warcraft III 1.26 $ wine Frozen\ Throne.exe or nimesh Warcraft III 1.26 $ wine Frozen\ Throne.exe -opengl # If the above doesn't work for you well.
One problem that I regularly face while switching windows is that the Alt key becomes sticky, causing signalling in the map whenever I click somewhere. Keep Calm, and press Alt once again and release. This will solve it.