There is a pretty well known bug with Dota2 under Linux on Intel graphics hardware. Essentially, models textures are missing as can be seen from the screenshot below:
I had at first tried installing lts-raring hardware enablement but that did not solve the problem and created new ones with my hardware. According to the bug-report, the problem was solved in Mesa 9.1.5 but lts-raring came with 9.1.4 only.
So, I downgraded back to lts-quantal and looked for another solution.
In the end, all I had to do was to add a 3rd party PPA to solve the problem:
# apt-add-repository ppa:glasen/intel-driver
# apt-get update
# apt-get -y dist-upgrade
This would install the Mesa 9.1.6 and Intel drivers, which solved the problem.
Now, I get to experience Dota2 in all its glory!