Remove tannins from tea without removing caffeine
In response to the answer posted by @James, I've run the following experiment:

Ingredients:


Black tea by "Tea Merchant", flavor = "Ship Ahoy"
Colorless and flavorless gelatin powder, bought from the baking isle in a supermarket.


Process:

1) I made 4 cups of tea, each with 1 tea spoon of tea leaves straight into the cup:  


Control cup, no addition
1/4 teaspoon of gelatin powder
1/2 teaspoon of gelatin powder
full teaspoon of gelatin powder


2)  I thoroughly stirred the tea and left the cups to sit for 4 minutes.

3)  Poured each cup through a strainer into a fresh receptacle to remove the tea leaves.  

4)  I left the tea to cool, such that I could take large gulps at a time in order to maximize tannin taste.

Results:

a) The colour was noticeably lighter the more gelatin was added.  Below is a photo, where each label indicates amount of gelatin in each cup (as fractions of a teaspoon, where left is full spoon, right is control).



b) A large gulp of the control tea was unpleasantly bitter.  

c) Each subsequent concentration removed more bitterness, indicating that a relatively large amount of gelatin is required to completely remove all tannins. 

d) 1/4 teaspoon was sufficient to remove the unpleasant sharpness

e) The full teaspoon removed most of the tanniny flavor, allowing the more subtle flavors to be tasted.

f) Although I find the full teaspoon left the tea a bit bland as well.  I prefer the 1/2 teaspoon ratio.

g) I am slightly disappointed that I did not create tea-jelly.  Will just have to add more.  For science.

h) I cannot comment on the effect of gelatin on the caffeine levels since I am quite caffeine insensitive.

Conclusion:

Gelatin definitely reduces tannin concentrations which removes the sharpness out of black teas.
I am unable to comment on caffeine levels from "feel", however, this paper models caffeine release from a gelatin matrix.  I only gave the paper a quick scan, and I find no mention of gelatin reacting with the caffeine in any way.



EDIT:

Over the course of the weekend I was able to feed a caffeine sensitive friend some tea.  Everyone was weirdly suspicious.

Process:

5)  I created two 1l batches containing 3 teaspoons of tea leaves, soaked for 10 minutes.  The control batch was left untouched and gelatin batch contained 10g of gelatin.

6)  Saturday morning at 10 am the subject was given the control tea.  He finished it by 10:45, and by 11 am his symptoms were:  Elevated heart rate (110 bpm compared to 60 at rest), headache, shakiness and a general feel of ill ease.  By 11:30 the symptoms seemed to have passed.  

7)  Sunday morning at 10 am the subject was given the gelatin tea.  To my great pleasure the tea had partially jellified.  The subject was not pleased.  He finished the tea by 10:45 and reported no symptoms at all, apart from feeling "a bit better".  It seems that the subject had somehow ingested some alcohol the previous evening, thereby bringing these results into question.

Results:

f)  The gelatin heavy tea didn't really taste that great.  The control was expectantly very bitter.

g)  We created tea-jelly!

h)  Based on the results from subject Friend, gelatin does in fact seem to lower the caffeine levels.  Although it may be possible that the jelly caused the caffeine to be released slower (or something).  Will have to repeat with much less gelatin, and again with much more gelatin.

i)  This would make a really good masters research project.