print "発刊日: " 
day = gets.chomp.to._i 

.to_iの間違いが原因ではと私個人は予想します
以下、コピペ

#encoding : utf-8
require 'date'

class BookInfo
  def initialize( title, author, page, publish_date )
    @title = title
    @author = author
    @page = page
    @publish_date = publish_date
  end

  attr_accessor :title, :author, :page, :publish_date
  def to_s
    "#{@title}, #{@author}, #{@page}, #{@publish_date}"
  end

  def toFormattedString( sep = "\n" )
    "書籍名: #{@title}#{sep}著書名: #{author}#{sep}ページ数: #{@page}ぺ―ジ#{sep}発刊日: #{@publish_date}#{sep}"
  end
end

class BookInfoManager
  def initialize
    @book_infos = {}
  end

  def setUp
   @book_infos["Yamada2005"]  = BookInfo.new(
   "実践アジャイル ソフトウェア開発法とプロジェクト管理",
   "山田 正樹",
    248,
  Date.new( 2005, 1, 25 ) )
   @book_infos["Ooba2006"] = BookInfo.new(
   "入門LEGO MINDSTORMS NXT レゴブロックで動くロボット",
   "大庭 慎一郎",
   164,
   Date.new( 2006, 12, 23 ) )
  end

  def addBookInfo
  book_info = BookInfo.new( "", "", 0, Date.new )
  print "\n"
  print "キー: "
  key = gets.chomp.encode("UTF-8")
  print "書籍名: "
  book_info.title = gets.chomp.encode("UTF-8")
  print "著者名: "
  book_info.author = gets.chomp.encode("UTF-8")
  print "ページ数: "
  book_info.page = gets.chomp.encode("UTF-8").tr("０-９","0-9").to_i
  print "発刊年: "
  year = gets.chomp.encode("UTF-8").tr("０-９","0-9").to_i
  print "発刊月: "
  month = gets.chomp.encode("UTF-8").tr("０-９","0-9").to_i
  print "発刊日: "
  day = gets.chomp.encode("UTF-8").tr("０-９","0-9").to_i
  book_info.publish_date = Date.new( year, month, day )
  @book_infos[key] = book_info
  end
  def listAllBookInfos
  puts "\n---------------------"
  @book_infos.each { |key, info|
  print info.toFormattedString
  puts "\n----------------------------"
  }
  end
  def run
    while true
      print "
      1.蔵書データの登録
      2.蔵書データの表示
      9.終了
      番号を選んでください(1,2,9) :"

     num = gets.chomp.encode("UTF-8").tr("０-９","0-9").to_s
     case
     when '1' == num
      addBookInfo
     when '2' == num
      listAllBookInfos
    when '9' == num
      break;
    else
    end
    end
  end
    book_info_manager = BookInfoManager.new
    book_info_manager.setUp
    book_info_manager.run
  end