It depends on the type of the error, but most errors fail quietly in Django templates. For example, you can pass a Python object to your template, and can reference its fields using syntax like {{ object.absolute_url }} if, for example, you tried to access a field that doesn't exist {{ object.doesnt_exist_field }} then it simply doesn't emit anything (as opposed to failing loudly).

There are a few errors in the template language that do fail loudly, and these fail when the templates are compiled (something you have the easy ability to test by simply loading the page once on a development server. Or by writing unit tests with the Django testing framework to make sure that all your urls are returning non-error response codes). Anything that has to do with the specific context of the page you are loading (the specific instance of an Object you are looking at, etc) may cause some ugliness (blank space where you wanted to print the value of a non-existant value), but the page will still load properly.

So, answering your question, there are a few types of syntax errors in the template language (loading a non-existent template library, improperly using tags or filters) that can cause errors (but all of those errors are easily testable). But the vast majority of errors will simply be ignored and normal functioning will continue.