Many shared hosting companies enable open_basedir by default. This is done for security purposes, since it limits the files that can be opened by PHP to a specific user.
An explanation of open_basedir can be found in the PHP manual:
When a script tries to open a file [...], the location of the file is checked. When the file is outside the specified directory-tree, PHP will refuse to open it.
When open_basedir is in effect, and you encounter an error, you will also see the following warning displayed:
open_basedir restriction in effect
The warning is unlikely to be what’s causing the actual issue, as Magento works with open_basedir restrictions. You will want to look past the warning at the actual error message being generated to resolve your issue.