There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of the Handbook of Algebraic Topology (this chapter was written by C.A. McGibbon), of a phantom map $\Omega^2S^5\to \mathbb{H}P^\infty$ which has this property.

My question is really simple to state: are there maps of spectra which are globally non-trivial, but become null when localized at any prime? Or, can anyone give a proof that this is impossible?

It's well known that there's an "arithmetic square" for reconstructing spectra from their rationalizations and their $p$-completions (cf. these notes), and I believe this might have some bearing on this situation?