Yes, resonance can definitely destabilize.
One example you can think of is destabilization of carbanions, which is something we often consider when trying to determine the relative acidities of some given species.
Consider the relative acidities of o-Methoxytoluene, m-Methoxytoluene and p-Methoxytoluene. Destabilization of their respective conjugate bases due to resonance plays an important role here, as mentioned in the renowned book Solomons|Frhyle|Snyder.
As clear in the attached figure, +R effect, where a substituent increases the electron density in a ring, destabilizes the conjugate base. Hence, the order of relative acidities here is: B > A > C
I encourage the reader to try and draw the resonance canonical forms for the three species mentioned, and realize why this happens for themselves.