Why does the sound pitch increase on every consecutive tick at the bottom of a filled cup of coffee?

I think you are observing "the hot chocolate effect" or something similar. See Crawford, Am. J. Phys. 50, 398 (1982). I have to confess I haven't read through the paper in enough detail to adequately summarize it.

Abstract:

The ’’hot chocolate effect’’ was investigated quantitatively, using water. If a tall glass cylinder is filled nearly completely with water and tapped on the bottom with a softened mallet one can detect the lowest longitudinal mode of the water column, for which the height of the water column is one‐quarter wavelength. If the cylinder is rapidly filled with hot tap water containing dissolved air the pitch of that mode may descend by nearly three octaves during the first few seconds as the air comes out of solution and forms bubbles. Then the pitch gradually rises as the bubbles float to the top. A simple theoretical expression for the pitch ratio is derived and compared with experiment. The agreement is good to within the 10% accuracy of the experiments.

I first noticed this in a hot cup of Horlicks, made with milk. I would stir in the powder vigorously, then tap the bottom of the cup with the spoon to check that all the powder had dissolved. Even two taps, one second apart is enough to detect the rising pitch. It continues rising and rising over the course of, perhaps, 20 seconds.

The interesting thing is that you can make the pitch drop again by stirring it up again. It seems that the pitch is directly related to the rate at which the milk is spinning.

I made this animation to explain this concept.

But let me answer some of your questions.

• works better with coffee than with tea (works hardly at all with tea) works better with cappuccino than with normal coffee

Since the effect is caused from air trapped in the powder, the reason why you won't always get the same effect, is because of how much air is trapped in the powder under production and how fast it is released when it's mixed with hot water.

• doesn't work with just cold water

Cold water won't dissolve the powder. At least not fast enough, and thus the air wont dissolve into the system. You can see if you pour in cold water on coffee powder, big chunks of coffee floats around.

• doesn't work on all types of cups, taller cups seem to work better

Not too sure about this one, but the equation that describes the frequency in the system is: $f = \frac{1}{4} \cdot \frac{v}{h}$ where v is the speed of propagation, and h is the height of the cup. I think it's just easier to distinguish this effect with a lower frequency (big cup) since the frequency is already quite high for cups already.