On my 4 year old white 13" MacBook (first revision, A1181, Core Duo) this started to occur for about 10 minutes after waking it from sleep. The trackpad button (which is a separate button on that model) would not "click" (so: was really pressed all the time). After the MacBook was heated up a bit things were fine though. And removing the battery fixed the problem right away. So it seemed there somehow was too much pressure on the trackpad while the machine was not heated up yet.
My current battery looks fine; an earlier one was swelling and looked a bit like the deformed one in Macbook Trackpad Not Working – Battery May Be To Blame but at that time I did not have the trackpad problems yet.
First, I partly fixed it by removing a little grey rubber sticker below the battery. That sticker was sitting right on the little round metal plate (which I guess is the actual button) where someone else put some additional material to apparently get more pressure in An Origami solution to the Apple MacBook TrackPad “squishy” Button Problem. I'm not sure if my MacBook had that sticker when I first purchased it; the sticker is not in the pictures in the ibloggedthis.com article, but I had my top case replaced some years after purchasing it.
The grey rubber sticker is partly visible on this ifixit.com picture:
After a few days, I noticed things were still not 100%. I guess it was a combination of dirt and the battery being in place. Some people report that dragging some folded piece of paper below the trackpad button might clean things. However, one cannot only clean underneath the button itself, but also in the opposite direction, between the button and the top case. The button is actually larger than one can see: the lower part of the button extends a bit to the left, right and front, underneath the top case. On my Mac, dragging a piece of paper a few mm underneath the button had no effect, but doing the same underneath the top case (while keeping the button pressed down) finally made it click like it was brand new.