Tinacos or the water tanks on the roof are indeed very simple. The problem is not the pump. It doesn't need a lot of pressure to fill. All that is required is that the water get there. The problem is probably the non-plumbers who installed it.
In the tank, there are two fittings and a lid. The water comes into the top through a float valve similar too, but simpler than a toilet stool-more like the floats used in livestock tanks on the farm. When the water reaches the designated level, the float raises and shuts off the water.
That brings us to step number 1. Climb up on the roof, open the lid and look inside the tank. There should be water. If not make sure the water is actually on. If it is, but is not getting to the tinaco, check the pipes and any valves starting in most cases with the float which could just be stopped up. It should more up and down easily.
At the bottom of the tank, there is an outlet that will be connected to the house plumbing. There must be a check valve in that line between the tinaco and the house. A check valve is a valve that allows flow in only one direction. If water starts to go through it the wrong way, it shuts off. It has a flapper like mechanism in it. Very simple. Its purpose to make sure that when there is pressure from the street or pump that no water enters into the tank through the bottom outlet because there is no shutoff.
That brings us to step number 2. Climb back up on the roof and make sure there is nothing clogging the bottom outlet and that the check valve is installed correctly. It should have an arrow on the outside indicating proper direction of flow. If it was installed backwards, you would have a full tank with nary a drop every draining out.
Normally, there is only one pipe that goes from the ground or house plumbing up the roof to the tank. When the water is on, water goes up through that pipe to fill the tinaco. When it is off, it come back down through it to run into the house. Somewhere in the vicinity of the tank, usually, the line is split so that one part goes into the top and the other connects to the bottom of the tank. The check valve must be after the split or it will never fill.
There is one more thing that is very, very important.
There must be another check valve between the house and the street. This prevents the water from your tinaco from entering the public water supply and essentially being a water tower for the street. It would be a kind gesture and supply water for your neighbors for a few minutes, but of course your tank would drain very, very quickly. Should that valve not be installed or not be functioning, you will actually have water for a short time from your tinaco when the water is off and it will fill normally but it will drain very quickly- perhaps in an hour or so.
Step 3 Climb back up on the roof and put the lid back on the tinaco. Like a dummy, you remebered it just as you finished putting the ladder away.