Depends. The question is one of takeoff performance mostly. I would do a W&B on the initial departure leg only, if my only weight changes on each leg are from fuel burn, and the subsequent legs are from airports with ample takeoff distance margin.
However, if an airport on my route is marginal from a takeoff distance perspective, my All Up Weight when taking off from that airport becomes critical and I would do a W&B adjustment to account for the reduced All Up Weight in determining if can get out of the strip in the first place. But I would be doing that estimation before I departed on the first leg anyway, and probably revisit it when at the airport in question to make sure my assumptions still applied.
Otherwise, if you just hopping around from 4000 ft airport to 460004000 ft airport, there's really no need. In fact as a general practice I only do detailed walkarounds and engine runups on the first flight of the day, with more of a cursory once-over on subsequent legs to make sure nothing happened to it while it was parked.