Circles will not tessellate. It's easy to imagine aligning a grid of circles where each circle touches adjacent circles at four points. The circle leaves a shape that is approximately a diamond-shaped quadrilateral with concave sides. It doesn't matter what size the circle is or if you fill the void with smaller circles. You always return to the same void. The best you can do is make it smaller.
Simple ovals or egg shapes that connect only two radii won't tessellate. The problem is the same as the circle. While it's possible to cover more area with an oval than it is with circles, voids are still left. Each oval only touches adjacent ovals in four points.
Where two curves bisect one another, it is called an intersection of articulation. Curved lines can be used in patterns that tessellate, but not if they don't have such articulations where one simple radius bisects another. Otherwise, you have the same problem with circles and ovals in tessellations but with a smaller radius of a shape that connects two or more radii.
While all squares can tessellate, some non-squares can tessellate and some can't. The name "tessellation" provides a hint to the shapes than won't tessellate. Its root meaning is "four," referring to squares. While many non-squares will not tessellate, a common trait in shapes that will is having three or more places where the edges of the shape bisect one another. The lines or "sides" of the shape may or may not have a curve, but the basic shape is multisided with intersecting lines.