Does my monitor's GTG response time matter if it's faster than my refresh rate?

Different sources of input lag add up. The limited refresh rate is but one source of latency, and the LCD's response time only adds to it. (Note that the GTG figure is intended to represent typical response time and does not reflect worst-case scenarios. Going from 0% to 100% white and vice versa often takes much longer than the 10% to 90% transition normally used for GTG measurements.)

As you've pointed out, a typical 60 Hz monitor can take up to 16.667ms to update itself. However, even after it updates its internal buffer, it still needs to send the display data to the panel, and the panel needs the specified GTG response time to react. With 8ms GTG, the display alone could add more than 24ms to input lag. On the other hand, if you have a 144Hz monitor with a GTG response time of 2ms, the lag contributed by the monitor is limited to 9ms.

Your display is not the only source of input lag. Your CPU, GPU, and even your mouse all contribute to input lag (hence gaming mice with 1000 Hz polling rate). This AnandTech article has an in-depth explanation of the topic.