That example will really help others to help you 
Using your example code I was able to continually reduce the code required to produce the problem, until I arrived at this minimal code:
shinyApp(
ui = dashboardPage(
dashboardHeader(),
dashboardSidebar(
sidebarMenu(
menuItem(
"menu", icon = NULL,
menuSubItem(
textAreaInput("text", "text"),
icon = NULL
)
)
)
),
dashboardBody()
),
server = function(input, output) {}
)
This shows us that the problem is not because of the select input. It looks like it's happening because the dashboard places the menusubitem inside a hyperlink. You can notice that by the fact that if you hover your mouse over the entire area, the cursor changes to a link cursor and if you're on chrome then in the bottom left corner you'll see the link.
To show this in an even clearer way, type menuSubItem("test") into the R console, and you'll see the output
<li>
<a href="#">
<i class="fa fa-angle-double-right"></i>
test
</a>
</li>
which clearly shows that whatever you pass to menuSubItem ends up inside a hyperlink (the a tag)
All of this was just to show you how I debugged this to try to find the source of the problem.
Below is my guess on why you're encountering this problem:
Looking at the documentation of menuSubItem(), it says
Menu items (and similarly, sub-items) should have a value for either href or tabName
Which is violated here. I definitely have seen many dashboards that have inputs inside of the sidebar as menu items, but it looks like (at least my understanding of it) this is not intended behaviour, and the menuitems are meant to only act as clickable links to specific sections in the page. If the menuitems are used as intended, rather than as inputs, then this issue would not come up.
That's only my theory, perhaps there is a standard way of including inputs as menu items, but it seems to me that it's not meant to be supported.
Hope this helps