Firstly, do not delete any of the page controls, as this will lead to lots of errors and a broken master page.

Instead, delete the original link for the Search Bar, for example.

    <asp:ContentPlaceHolder id="PlaceHolderSearchArea" runat="server">
    <SharePoint:DelegateControl runat="server" ControlId="SmallSearchInputBox"/>
    </asp:ContentPlaceHolder>

And create a new section, just before the </form> tag:

  <!--- --- ---  Hidden Placeholders --- --- --->
  <asp:Panel visible="false" runat="server">
   <!-- Place unused Content Placeholders here -->
    <asp:ContentPlaceHolder id="PlaceHolderSearchArea" runat="server"/>
  </asp:Panel>

Anything in this hidden section will disappear from view but not break the page!