public bool RemoveItem( 
   IProjectItem item
)

/// Remove a folder connection from a project; the folder stored on the local disk 
/// or the network is not deleted
FolderConnectionProjectItem folderToRemove = Project.Current.GetItems<FolderConnectionProjectItem>().FirstOrDefault(folder => folder.Name.Equals("PlantSpecies"));
if (folderToRemove != null)
  Project.Current.RemoveItem(folderToRemove as IProjectItem);

/// Remove a map from a project; the map is deleted
IProjectItem mapToRemove = Project.Current.GetItems<MapProjectItem>().FirstOrDefault(map => map.Name.Equals("OldStreetRoutes"));
var removedMapProjectItem = await QueuedTask.Run(
         () => Project.Current.RemoveItem(mapToRemove));

string newGDDItemPath = @"C:\path\ArcGIS\Project\NewLocation.gdb";
string oldGDBItemPath = @"C:\Path\Project\OldLocation.gdb";

//Create a new GDB item and add it to the project
var newGDBItem = ItemFactory.Instance.Create(newGDDItemPath) as IProjectItem;
if (newGDBItem == null)
  return;
var success = Project.Current.AddItem(newGDBItem);
//make the newly added GDB item the default
if (success)
  Project.Current.SetDefaultGeoDatabasePath(newGDDItemPath);
//Now remove the old item
var oldGDBItem = Project.Current.GetItems<Item>().FirstOrDefault(i => i.Path == oldGDBItemPath) as IProjectItem;
if (oldGDBItem == null)
  return;
var removeSuccess = Project.Current.RemoveItem(oldGDBItem);

// Remove a folder connection from a project; the folder stored on the local disk or the network is not deleted
FolderConnectionProjectItem folderToRemove = Project.Current.GetItems<FolderConnectionProjectItem>().FirstOrDefault(myfolder => myfolder.Name.Equals("PlantSpecies"));
if (folderToRemove != null)
  Project.Current.RemoveItem(folderToRemove as IProjectItem);

//This example deletes a layout in a project after finding it by name.

//Added references
using ArcGIS.Desktop.Core; 
using ArcGIS.Desktop.Layouts;

public class DeleteLayoutExample
{
  public static Task<bool> DeleteLayoutAsync(string LayoutName)
  {
    //Reference a layoutitem in a project by name
    LayoutProjectItem layoutItem = Project.Current.GetItems<LayoutProjectItem>().FirstOrDefault(item => item.Name.Equals(LayoutName));

    //Check for layoutItem
    if (layoutItem == null)
      return Task.FromResult<bool>(false);

    //Delete the layout from the project
    return Task.FromResult<bool>(Project.Current.RemoveItem(layoutItem));
  }
}

//Remove a layout project item.

//Remove the layout fro the project
await QueuedTask.Run(() => Project.Current.RemoveItem(layoutItem));

//Note: Call within QueuedTask.Run()
//Reference a reportitem in a project by name
ReportProjectItem reportItem = Project.Current.GetItems<ReportProjectItem>().FirstOrDefault(item => item.Name.Equals(reportName));

//Check for report item
if (reportItem == null)
  return Task.FromResult<bool>(false);

//Delete the report from the project
return Task.FromResult<bool>(Project.Current.RemoveItem(reportItem));