This is a very good topic. I say that. Because when I am drowning Minnows around structure. If I go weight free i tend to get hung up a lot more that when I put a split shot on. but back to the question. I try to go as light as possible.
Been there...when I need to fish real close beside heavy cover with a minnow, I tend to use a really heavy lead weight about an inch above the hook eye...the old style wrap around the line type...this allows me to fish right beside the structure without allowing the minnow to drag the hook into the brush. Basically, by using a heavy weight only an inch above the hook eye (as opposed to three inches above), I can drop right beside (3 inches) the heavy cover without fear of the minnow constantly snagging me up...theory still holds though, as you use the lightest weight you can for the type of fishing you are doing...this particular situation requires a heavier weight, similar to the drop shot rig that duck jerky mentioned, you could go with a heavier weight, but you don’t need to...so don’t.
Now, going weight free close to heavy cover is similar to using 4lb line in heavy cover...you can do it, but no sense in it...