Let $f\colon X\to Y$ be a continuous function which is not null-homotopic. This doesn't necessarily mean the induced map on singular cohomology is non-zero: for example if $c\colon S^{2n+1}\to BO(2n+1)$ classifies the tangent bundle, then the induced map on cohomology is trivial (since the tangent bundle is stably trivial and $H^*(BO(2n+1))$ only contains stable characteristic classes) but this tangent bundle is typically not trivial and so $c$ would not be null-homotopic.
If $f$ is not null-homotopic is there always a cohomology theory $h_f$ such that $f^*\colon h_f^*(Y) \to h_f^*(X)$ is not zero? Is there a cohomology theory which can detect ALL maps which are not null-homotpic?