The script execution environment of a Content script differs from the page. Consequently, runtime errors in a web page will generally not cause any problems in content script, and vice versa.
It's however possible that the content script just stops working because the expected conditions are not met (any more). For instance, an element was expected to exist in the document, but not found. The reverse can also be true: The content script modifies the page to such extend that the page doesn't function any more.
The Content security policy of the page does affect content scripts though. This can lead to unexpected results, such as blocked cross-origin requests (example).