What are the risks of running 'sudo pip'?

When you run pip with sudo, you run setup.py with sudo. In other words, you run arbitrary Python code from the Internet as root. If someone puts up a malicious project on PyPI and you install it, you give an attacker root access to your machine. Prior to some recent fixes to pip and PyPI, an attacker could also run a man in the middle attack to inject their code when you download a trustworthy project.

Besides obvious security risks (which I think are in fact low when you install software you know) brought in other answers there is another reason. Python that comes with the system is part of this system and when you want to manage system you use tools designated for system maintenance like package manager in case of installing/upgrading/uninstalling software. When you start to modify system's software with third party tools (pip in this instance) then you have no guarantee about the state of your system. Yet another reason is that sudo can bring you problems you wouldn't have a chance or have a very small chance to have otherwise. See for example Mismatch between sys.executable and sys.version in Python

Distros are aware of this problem and try to mitigate it. For example Fedora – Making sudo pip safe and Debian – dist-packages instead of site-packages.

Using pip that way means you trust it to the level you allow it to make anything to your system. Not only pip, but also any code it will download and execute from sources you may not trust and that can be malicious.

And pip doesn't need all that privileges, only the write access to specific files and directories. If you can't use your system's package manager and do not want to go the virtual environment way, you may create a specific user that has write privilege to the python installation directory and use it for pip. That way you better control what can pip do and not do. And you can use sudo -u for that!