Below are the two different solutions if you want to display dynamic dates on header in Shiny.
This code will be helpful if you need to use shinydashboard package
library(shiny)
library(shinydashboard)
header <- dashboardHeader(
  title = "dynamicDates",
  tags$li(class = "dropdown", tags$a(HTML(paste(uiOutput("Refresh1"))))))
body <- dashboardBody()
sidebar <- dashboardSidebar()
ui <- dashboardPage(header, sidebar, body)
server <- function(input, output) {
  output$Refresh1 <- renderText({
    toString(format(Sys.Date(), format = "%A  %d %b %Y"))
  })
}
shinyApp(ui, server)
In this code i'm not using shinydashboard package. With plain shiny functions in combination of few HTML tags we can customize as per need.
library(shiny)
ui <- fluidPage( 
  titlePanel("", windowTitle = "Dynamic Dates"),
  titlePanel(title = tags$div(img(src = "https://www.rstudio.com/wp-content/uploads/2014/04/shiny.png", width = 125, height = 115, align = "left"))),
  titlePanel(title = tags$div(class = "header" , tags$p("Dynamic", tags$b(" Dates"),style = "text-align: center; color:navy;"), style = "text-align: center; color:navy;")),
  titlePanel(title = tags$div(uiOutput("dynamicDate"), align = 'right')))
server <- function(input, output) {
  output$dynamicDate <- renderUI(toString(format(Sys.Date(), format = "%A  %d %b %Y")))
  }
shinyApp(ui, server)