Maybe a little since the packages are compiled for the right cpu. I doubt you would notice any difference though.
There are differences especially when not running synthetic benchmarks (real-world applications). But that's not due to ARMv6 vs. ARMv7 compiler settings (in fact Raspbian guys did a really good job to optimize their ARMv6 builds as much as possible, I checked several binaries already where Raspbian binary is faster than Debian armhf counterpart) but due to the more important stuff like cpufreq governor tuning and IO scheduler and priority settings.
Not directly comparable but that's the old Rasbian based OMV image for Raspberries tested with a pretty fast SSD:
And these are comparison numbers of DietPi (Raspbian based) and the new image made with a 2.5" HDD: New approach for Raspberry Pi OMV images
HDD vs SSD strangely makes a difference since different bottlenecks add up so the new image combined with the same SSD as in the first test above will show superiour or equal numbers in every area. But that doesn't matter that much since Raspberries are too slow anyway (and if people look only at backup times, then good luck for them when they realize that restore times are important, do the math yourself how long it takes to pull 500GB off an Raspberry Pi at lousy speeds around 5MB/s compared to 50MB/s with a good ARM board, that's few hours vs. a whole day).
Current Raspbian is based on Stretch already so maybe the usual way upgrading to OMV4 works. It's easy to test (with a cloned SD card before) so I really wonder why affected users ask questions instead of providing insights