Zinc usually won't completely quench the luminescence of a bound ligand. E.g., compare the fluorescence yield of a porphyrin bound to zinc (3.3%) vs that of a porphyrin bound to iron (<<0.1%). Do note that the zinc does lead to some quenching. The fluorescence yield of free base (not bound to anything) porphyrin is about 13%. The quenching in the case of zinc is predominantly due to spin-orbit coupling (heavy atom effect), which increases efficiency of excited triplet state generation.
I would say that in most cases a luminescent ligand is going to be fairly rigid already (it has to be, to have a decent fluorescene yield), so the effect of increasing rigidly when binding to a metal on the fluorescence yield will be small, but I could imagine scenarios where this occurs. I could also imagine scenarios where the opposite would occur (binding of metal puts ligand in unfavorable conformation for fluorescence). These effects would have to be taken on a case by case basis - I think it's hard to speak generally. Keep in mind also in your case of interest, the luminescence comes from the bound lanthanide ion, not the ligand itself, so rigidity of the ligand wouldn't matter so much except for influencing the excited state lifetime of the ligand, which would increase the likelihood of energy transfer to the lanthanide.
Ultimately, it all comes down to relative rates. If your rate of energy transfer to the lanthanide is extremely fast, then it will remain competitive even with a nearby metal ion that can facilitate other types of excited-state relaxation. If your rate of energy transfer is slow, then the effect of a nearby metal ion would be comparatively large. It's been quite a while since I read any literature on lanthanide fluorescence sensitization from bound antennae, so I'm not sure off the top of my head what kind of rates we're dealing with, and what kinds of factors are necessarily important to increase energy transfer efficiency. But to answer your question: yes, in general a closed shell metal ion like zinc will quench the luminescence far less than an open shell metal ion.