I can do it in four lines. Start Line #1 at the top of the upper left dot. It angles down just a little from the horizontal, passes through the dots on the top row, and crosses the upper right dot at the *bottom* of the dot. Line #1 continues out to the left until it is halfway between the top and the second row of dots. Line 2 starts at the end of line 1, out to the left, and crosses the second row of dots from upper right to lower left. Line 3 picks up out to the right, is parallel to Line #1, and crosses the third row from upper right to lower left. Line #4 is parallel to line #2, and crosses the last row from upper right to lower left. Of course, we can do the same vertically, and that creates a giant letter "W" going through all the dots.
If you let me roll the paper into a cylinder of the right circumference, I can do it all in one line. The line starts as #1 above and keeps going around the cylinder, crossing the second row from upper left to lower right, then around a again a third time and a fourth time.
If anyone can do it on a flat pieces of paper in 3 lines, I'd like to know how.
By the way - I assume we're not allowed to move the dots. If we are, let's just put them all in a pile, and do it in zero lines!