To use the tree tap, right click on the orange-yellow blob on a Rubber Tree whilst holding a tree tap to extract the resin. The blob will disappear and sticky resin will appear in your inventory. This process will give you between 1-3 sticky resin.
Please note that there will be a small black hole on that part of the tree most of the time after this process. If you continue right-clicking on it, it will eventually disappear but can sometimes give you more sticky resin. But this will deplete your tree tap very quickly so be aware of your tree tap's health in your hotbar. By using the tree tap multiple times on the tree, it will eventually stop the tree from regenerating its sticky resin so please bare this in mind.
They are very weak tools and will run out very quickly so do not waste them unnecessarily. It is recommended that you eventually upgrade to an electric tree tap (below) for they do not run out but use EU instead. They can be charged in a BatBox or using a BatPack. An MFE or an MFSU may be used for charging this too.
Smelt the sticky resin in a furnace to gain 1 piece of Rubber.
Extract the sticky resin in an Extractor to gain 3 pieces of Rubber.
Extracting sticky resin with the treetap gives you the "Sticky Situation" achievement.
Although, once you have an extractor, it is recommended that you stop doing this and look for a RedPower Rubber Tree and destroy its 3x3 trunk and extract the wood in an extractor. Each piece of wood is worth 1 rubber once extracted. The RP Rubber Tree cannot be confused with a regular tree because it is absolutely massive. This is what one looks like: