There doesn't seem to be a study that focus solely on how people feel about when the amount of unread messages creeps ever higher. I'm sure there are studies in psychology on how amount of unattended/unfinished tasks affect on person's emotions. Probably they have something to do with motivation and what motivates us.
There is a somewhat classic study by Whittaker and Sidner on email overload: Email overload: exploring personal information management of email.
We demonstrate that email overload creates problems for personal information management: users often have cluttered inboxes containing hundreds of messages, including outstanding tasks, partially read documents and conversational threads. Furthermore, user attempts to rationalise their inboxes by filing are often unsuccessful, with the consequence that important messages get overlooked, or "lost" in archives.
The study suggests that people have different kinds of methods to cope with the amount of messages (emails) received. Unread messages are sometimes used to remind users on something that needs to be done.
At least two studies exists that have looked into if the results of Whittaker and Sidner still holds:
From Grevet et al:
This suggests a possible connection between feeling unorganized and having many unread emails. Some of our participants offered their sentiments about their unread count when asked about how they felt about the number of emails in their inbox. They describe feeling of being overwhelmed by these unread messages[.]
We do not find an indication that a large number of [unread] emails in the inbox corresponds to a strong sense of disorganization. Most of our participants with large inboxes (over 1001 emails in their inbox) feel fine or indifferent about the number of emails in their inbox[.]
Rather than categorizing people based on number of emails in their inbox, we could look at whether users can be categorized based on the number of unread emails in their account and strategies they use to keep this number low. We
saw that our participants with high unread counts felt the most overwhelmed, future email studies could focus on this category of users in particular.
So there are still things to study in the field of unread messages and their effect on users.
Attending to Email by Hanrahan, Pérez-Quiñones and Martin might answer some questions. From abstract:
We found that participants attended to email primarily based on notifications, instead of the number of unread messages in their inbox.
Number of unread messages might have an effect on email triaging, when user decides when and where to start triaging messages. From Beyond "From" and "Received": Exploring the Dynamics of Email Triage by Neustaedter, Brush and Smith:
As discussed, email triage involves deciding what to do with "unhandled email," yet prior to the study it was not clear what this entailed. We found in our interviews with [high volume] triagers that they do not often look at the contents of folders when triaging; rather, triaging is primarily performed on unread emails in the inbox, with some people also looking at read items.
Google Scholar search on "unread messages", without patents, can also yield some useful results.