Excretion does remove thermal energy from the body, but it also removes mass, and as a result your heat to mass ratio doesn’t change in any meaningful way.

“Keeping it in” would technically make your body temperature slower to change (Up or down) because there is slightly more mass to heat or cool. Excreting would technically make your body temperature slightly more susceptible to change, again because there is less mass to heat or cool. But really, those changes are inconsequential.

The actual cooling would occur on intake, not excretion. When you drink cold water, your body heat will dissipate into that water until the temperatures match, resulting in a slight reduction in temperature.

So in summary, excretion itself does nothing to cool you down, even though it’s taking thermal energy away, but the entire cycle of drinking cold water, heating it in the body, and then excreting it would reduce body temperature ever so slightly.