server.R

library(shiny)
library(magrittr)

shinyServer(function(input, output, session) {
  
  #__________________________________________________________________________________
  # The main named list that will be used in many other tasks.
  listN <- reactiveValues()
  makeReactiveBinding("listN")
  
  output$uiAdded <- renderUI({
    selectInput('added',
                'List of combinations', 
                choices = names(listN),
                multiple = TRUE,
                selectize = FALSE)
  })
  
  #__________________________________________________________________________________
  observe({
    # Trigger Add actions
    input$actionBtnAdd
    
    isolate({
      new_selections <- input$add
      new_selections_name <- new_selections %>% paste(collapse = " X ")
      
      if(new_selections_name != "")
        listN[[new_selections_name]] <- new_selections
    })
  })
  
  #__________________________________________________________________________________
  observe({
    # Trigger Remove actions
    input$actionBtnAdded
    
    isolate({
      # identify the items selected by user to be removed
      selections_to_remove <- input$added
      
      listNAsList <- list()
      
      # Here I manipulate 'listN' by converting it to list (using 'reactiveValuesToList()')
      if(!is.null(selections_to_remove)) {
        # To list so it can have their items removed
        listNAsList <- reactiveValuesToList(listN)
        
        # remove those items selected in input$added
        sapply(selections_to_remove, function(x) {
          listNAsList[[x]] <<- NULL
          return(NULL)
        }) %>% invisible
      }
      
      # Now, I'll try to update 'listN' by recreating it from scratch with only the remaining items in listNAsList object.
      list_names_for_choices <- names(listNAsList)
      if(is.null(list_names_for_choices)) { # if there is no items left...
        
        # ...an empty list is returned (without options, as desired)
        #listN <- reactiveValues()
        
        # Here is a workaround to clear the 'added' list. NULL or character(0) would keep selectInput unchanged.
        list_names_for_choices <- ""
      } else { # if there is some remaining items on 'added' input list...
        
        # ...update listN with it.
        listN <<- do.call(reactiveValues, listNAsList)
      }
    })
  })
  
  #__________________________________________________________________________________
  # debug output to show the listN content.
  output$debug <- renderPrint({
    listN %>% reactiveValuesToList
  })
})

ui.R

library(shiny)

shinyUI(fluidPage(
  titlePanel("reactiveValue Issue"),
  tags$p("I need to be able to add and remove items from a reactiveValues object, but its items is annoyingly persistent. Please, help me to figure out WHY."),
  fluidRow(
    column(2,
           selectInput('add', 
                       'Multiple Choices', 
                       LETTERS[1:10], 
                       multiple = TRUE, 
                       selectize = TRUE),
           actionButton('actionBtnAdd',
                        'Add',
                        icon("plus"))
    ),
    column(2,
           uiOutput("uiAdded"),
           actionButton('actionBtnAdded',
                        'Remove',
                        icon("minus"))
    ),
    column(6, offset = 1,
      h4('debug area: "listN" content, the object of interest'),
      verbatimTextOutput('debug')       
    )
  )
))