What I did is when a tile goes ofscreen on the right (just an example) that the same tile should be drawn left from the tile which is most left to fill up the empty space. I used the same trick for moving objects so that if a monster goes out of the screen on one side I just draw it twice temporary on the other side.
Here is a tiny example which only checks if the object goes ofscreen on the right and if so it draws it twice :
Strict
Global width:Int=640
Global height:Int=480
Global posX:Int=320
Global posY:Int=240
Graphics width,height,0
While Not KeyHit(Key_Escape)
Cls
posX = posX + 1
If posX > width + 50 Then posX = posX - width
DrawOval posX - 50,posY - 50,100,100
If posX > width - 50
DrawOval posX - 50 - width,posY - 50,100,100
EndIf
Flip
Wend
End
I think it is faster than trying to capture every pixel which goes ofscreen on one side and draw them on the other side.