It's interesting that because another name for tannins are tannic acids, you would think pH would drop significantly with lots of tannins, and rise without them.
Soon after the rains started here, and my 180 gal tank became tea colored , I expected pH to drop, so did a few pH tests.
Here are the results.

Below ...Compare to a pH test from back in January, before the rains started, and the tank had few if any significant tannins dripping in

Below ....a shot from the same day in January showing the difference in water color from the one taken a few days ago, a few posts above.

There are factors that determine whether of not pH rises or falls.
Lack of pH drop may have to do with the waters buffering capacity.
If your waters buffering capacity (AKA alkalinity) is low, your pH may drop like a rock.
If your tap water alkalinity is high, the calcium in it prevents pH from falling.
Below is a test done about 3 years ago of my island tap water.

The 5th row down is Total Alkalinity and reads @ 100+ ppm of Alkalinity.
This is a fairly strong buffering capacity.
When i was living in Milwaukee, the tap water Alkalinity was slightly lower at around 90ppm, so easier to manipulate pH.