There are two likely contributing factors here. What you're seeing is rapid release of carbon dioxide from solution, indicating that its solubility has decreased and that the rate of re-equilibrating to that new lower solubility is quite fast. The solubility of carbon dioxide is a factor of
1) the partial pressure of CO2 in the air above the liquid (higher pressure forces more into solution),
2) the pH of the liquid (because higher pH leads to formation of more $\ce{HCO3-}$ and $\ce{CO3^2-}$ thus increasing the total CO2-related species in solution), and
3) temperature of the solution (like most gases, CO2 decreases in solubility as temp increases).
In your case, (1) and (3) are probably most important. [EDIT: I overlooked that the coffee was cold, so (3) is actually not likely to be relevant] The can before opening is pressurized with a high CO2 atmosphere, so it is overloaded with CO2 relative to just about any ambient condition. In order to equilibrate with ambient conditions, it needs to lose some CO2. On the other hand, you've diluted it into the coffee, so there is a slight offsetting effect there. The coffee is slightly acidic, but is not likely to be very well buffered, and probably is not much different than diluting with water.
In addition to these thermodynamic effects that change the equilibrium dissolved CO2 concentration, there is also the kinetic issue of how quickly the CO2 escapes in order to return to equilibrium. The CO2 can only leave solution at the boundary points where the liquid meets air. The stirring effect of pouring the liquid (even into an empty glass) creates more surface area of the liquid/air boundary and also keeps the CO2 well distributed throughout the solution so that the areas near the boundary are continuously resupplied with CO2 as it is lost to the air.
To determine if coffee specifically has any effect, I would try the same thing with water instead of coffee. I suspect the result will be very similar.
UPDATE based on new data:
I was wrong about the coffee itself not being important. The coffee mixture clearly foams more than the tea. There are two possibilities I can think of (not mutually exclusive):
1) the rate of CO2 evolution is greater in coffee. As the other answer postulates, that could be due to faster nucleation by particles in the coffee or, if the pH goes up very quickly, buffering by acids in the coffee once the pH reaches 5 or so (from the Coke starting point ~ pH 2).
2) the same amount of CO2 is released, but the coffee mixture foams more. Studies on sparkling wines such as the one described here indicate that proteins, especially ones with carbohydrates attached, contribute to foaming by acting as surfactants (that means like a detergent molecule, essentially making soap bubbles). It is likely that coffee contains similar molecules. I now think this is the likely explanation, though I also like the idea that there are more nucleation sites from the dissolved solids.
The way to test some of these hypotheses would be to filter the coffee with varying molecular weight cut-off filters, but I suspect you don't have access to that sort of equipment?
Above is after the coffee cooled.
