Doesn’t the Laplace–Beltrami operator on a curve simplify to the ordinary 1d Laplace operator if you parameterize the curve by arc length?
But things may be more complicated if you are not talking about a mathematical curve, but a physical wire (of nonzero diameter) that you want to approximate by a 1d equation along the curve. As you bend a physical wire, I suspect that it changes the corresponding 1d approximate heat-flow equation depending on the curvature. But this is not about differential geometry, it is about the difference in Laplace solutions for a cylinder vs. a torus, as well as distortions in the material properties by strain (and there are other effects in nanoscale wires) — I’m guessing that there are corrections of order (d/R)^2, where d is the wire diameter and R is the radius of curvature (squared because it shouldn’t depend on the sign of R?).