The short answer is "yes" you should check, however this is not as arduous as it might appear at first. Your initial convergence calculations will be fairly exhaustive, sweeping over a large range of basis set size, Brillouin zone sampling etc. but when you change exchange-correlation functional you only need to check that your previously determined parameters were sufficient.
As an example, if I were to use a plane-wave pseudopotential DFT program I might start with LDA and converge the plane-wave cut-off by looking at cut-off energies of 300 eV, 400 eV, 500 eV... 1200 eV, and perhaps decide to use 600 eV. If I then change to PBE, I would start at 600 eV and check that this is reasonable by either comparing dE_total/dlog(E_cutoff) with the LDA one, or looking at 700 eV and perhaps 800 eV (comparing the trend to the corresponding trend with LDA). This is much quicker and easier than the original convergence check, since we know roughly what we expect the answer to be and how everything should vary.
The natural follow-on question is why the convergence might depend on the choice of exchange-correlation functional. The principal answer to this is the self-interaction error: the Hartree term in the Kohn-Sham equations is the Coulomb repulsion between the electron density at a point and the electron density at another point, but at each of these points some of this electron density was due to the same particle, which means that a particle repels itself! This is called self-interaction, and is completely incorrect. The "true" exchange-correlation functional would remove this spurious self-interaction but, since we don't have this wondrous functional at present, we must make do with approximations to it. These approximations vary considerably as to how much they mitigate the effect of self-interaction.
If we focus on the two main considerations of basis set size and Brillouin zone/k-point sampling (for periodic systems) then we can see what difference we expect from the self-interaction error.
The appropriate basis set size usually depends primarily on the states near the nuclei. A large self-interaction leads to spurious delocalisation of states with high electron density, particularly "shallow core" states such as atomic d- and f-states. This can change the size of basis needed to describe the states accurately. If you are using pseudopotentials, then there is an additional effect: when you change the functional you also change the pseudopotential, and this may require more basis sets. This is particularly true in my experience if you're using functionals with some Fock exchange, as the pseudopotential generation in this case is rather difficult and the resultant potentials tend to be rather hard.
The Brillouin zone sampling depends on the bonding in your system so you may not expect this to be strongly dependent on the exchange-correlation functional, but this is not always true. One of the clearest examples might be transition metal oxides, e.g. NiO, which is strongly affected by the self-interaction error in the Ni d-states. LDA predicts that NiO is a non-magnetic metal (due to self-interaction error), so an exhaustive convergence check would conclude you need a high k-point sampling, but do not need to worry about spin (collinear or non-collinear); however, changing to PBE removes just enough of the self-interaction to open up a small band gap and make NiO an antiferromagnetic insulator, meaning that you do not need as high a k-point sampling but you do need to consider the spin density (at least at the collinear level). The antiferromagnetic insulating state would also be obtained with a modest Hubbard U, even with LDA, on the Ni d-states, and/or a proportion of Fock or screened-Fock exchange (which removes this self-interaction directly).
