A good tv series is addictive, a good film makes people cry. The emotions people experience in a film might be transferable into serious gaming (games that serve a purpose), for instance "a loosing weight game/ calorie tracker". Might it be possible to transfer the techniques a script editor/ dramaturge/ story teller applies to a web application? Is there already an example or a theory?
A web application does need a storyboard. This is less about the entertainment factor, and more about the flow of a process through the application. The "story" is more the user journey through the processes - there will often be a few, and a few subplots too.
The right sort of drama for a web app is one one that is completely predictable, easy to follow, and with no twists. That is as hard to do as a complex dramatic script it would seem, as far too many applications have unexpected plot twists in them.
And applications that make users cry are generally a mistake. I have seen users crying and it is generally a bad move.
If you want to create a serious game, then you definitely need some kind of story. This narrative framing can be expressed by visuals, text, and/or interaction.
A good introduction for game-based learning applications is Ang+Zaphiris (2005): Developing enjoyable second language learning software software tools: A computer game paradigm. It distinguishes the perspectives of Ludology (the "fun" aspect) from Narratology (the "story" aspect), analysing "typical" games first, then applying it to language learning.
But as others have answered, the game aspect can be more or less useful, and can be used to different extents. Mailchimp, for example, provides an interesting, informal experience, yet doesn't make sending mails a "game".
The purpose of a good film or tv series is to entertain. Usually the purpose of a web application is to perform a task of some kind. Unless the goal of your app is to entertain, you may want to rethink combining aspects with very different purposes.
For me a good app is one that gets out of my way and lets me do what I need as simply as possible. Adding drama to that would just annoy me and encourage me to find another app to use.
Roo, if your intent is to build a web-based game, then absolutely have drama as part of the game itself. For example, the recent French production 'Hard Rain' is well known for its drama and depth of storyline, following multiple characters through a complex situation where the gamer's choices have real consequences, including life and death of the main characters.
If your goal is to provide drama in a similar setting then that sounds both interesting and valuable to the further development of web gaming. However, interfaces outside the game itself (such as settings, account, etc.) should always be predictable and easy to use. As pointed out in Schroedinger's Cat's answer, you never want to make users cry out of anything except emotion for the game itself.