The three utilities problem is a disguised version of asking whether $K_{3,3}$ is a planar graph. It's not, but $K_{3,3}$ *is* regularly embeddable in the torus -- if given one "handle" attached to the plane, you can attach the last house/utility pair. The general question is "how many handles do you need" for various numbers of utilities and houses. This is Turán's brick factory problem and is open (upper and lower bounds are known).

Brouwer's fixed point theorem tells me that if I start with a flat piece of paper, crumple it up, and drop it back where it started, at least one crumpled point projects (perpendicularly to the original sheet) back to its initial position. (The linked version of this example attributed to Brouwer is better stated.) Similarly, the Borsuk-Ulam theorem leads to the non-technical result that at every instant, there is a pair of antipodal points of Earth with identical temperatures and barometric pressures (or any other pair of continuous scalar properties you like).

The hairy ball theorem states (very informally) you can't comb a hairy ball without a cowlick. But you can comb a hairy torus without one. So there is a boring automated way to paint a torus, but not so much for a sphere.

There's always More Knot Theory...
You can demonstrate tying a knot in string without letting go of the ends during the tying (description and illustration). This gives an overhand knot, which is a chiral knot. You can tie its reflection by reversing the helicity of the starting state of the arms. Interestingly, not all knots are chiral; there are amphichiral knots.