My toilet tank does not fill back up with water after I flush it. I turned off the valve, disconnected the hose, and turned the valve back on to see if there might be a clog, but there was none. Also, there is a trickling dripping noise coming from inside the tank, but there is now water leaking from anywhere. What could be causing these problems?

I don't know what kind of a fill valve you have in your tank so I will just talk about things in general. First, does the float fall down after you flush the toilet? If it does that is what should let new water fill your tank, so if it is falling and no water comes into the tank there must be as stoppage in the valve itself. Therefore I would turn off the water supply and sponge all the water out of the tank. I would then take the fill valve out of the tank and look in the bottom of it to see if you can see something that would cause the water not to go through. If you don't see anything I would take the valve apart to see if something is in the upper part of the valve, if you don't find anything I would re-install the valve back in the tank, reconnect the water and with the valve still apart, that is with the top off, turn the water on very slowly with a glass turned upside down over the valve opening, so that if water does come out it will be turned back into the tank and not squirted to the ceiling. If no water comes out I would buy a new valve and replace the old one.

There is one valve that has a lever on the side of it that will not let the float fall unless the tank handle is tripped. If you have this you can see that you handle should lift it.

Now about the dripping noise, I think this is caused by the ball in the bottom of the tank not sealing good. You may have a ball with a wire screwed into the top of it or you may have a flapper that covers the hole in the tank, replace which ever one you have with the same kind and that should stop the dripping. If this post helps you please let me know.

Sounds like rubber flapper isn't sealing properly. If there is water coming into tank after you flush---------- RUBBER FLAPPER IS YOUR BAD-BOY :)

Is your fill valve one that has a ball on it or a single stalk? How old is the fill valve? If it is over 5 years old just get a fluidmaster 400 and replace it. If it is a fluidmaster and is not orange, the seal has swollen and needs to be replaced.