First we need to find out why the tank pH is higher than the tap, and there are two possible reasons.
When you tested the tap, did you let the water sit out overnight? Depending upon the amount of dissolved CO2 in the water, it will affect a pH test. Letting a glass of water sit for 24 hours (overnight should be sufficient) and then test will ascertain the more accurate pH, and it may be higher if this method was not used previously.
Second possibility depending upon the above is that something calcareous in the tank is dissolving and raising hardness and pH [the two are closely linked]. This could be rock, gravel or sand made from limestone, dolomite, marble, lavarock, or coral.
Attempts to lower pH often fail because pH is linked to the mineral hardness and a high bicarbonate hardness (KH or Alkalinity) will act as a buffer to maintain the pH where it is out of the tap. You can read more about this in my article: Water Hardness and pH in the Freshwater Aquarium
That will also contain a brief summary of how to lower pH, but before attempting that, post your answers to the above issues. Also, if you could give me the GH and KH of the tap water (find this out from the water supply folks, many have a website with data posted). All this will tell us what to do. Also, what fish are in the tank?
Briefly to your two questions, yes live plants help stabilize pH but not lower it. And wood will lower pH but minimally, and here again the KH is involved.