Changing the value of one affects the appearance of the other, hence it makes a difference whether you change the tint value first and then move on to the temperature, or the other way around.
They are independent, the temperature slider affects the color temperature, which is effectively blue-yellow, while the tint affects the green-magenta axis.
In Adobe Camera Raw, the temperature slider is before the tint slider, probably for a reason, so I think the best strategy is to adjust them top to bottom, and this is the advice given in many books on the subject.
Personally, I prefer to adjust temperature first, and then tint. It is just much easier for me to adjust warm-cold (yellow-blue) balance, then get the right magenta-green. So my tint adjustemnts tend to be smaller, and I often look for just particular “known” color like skin tone.
I believe there are some reasons to do temperature first:
- human eye is very sensitive to tonal difference in blue
- blue-yellow is the strongest color contrast (probably high sensitivity to yellow wavelength vs low sensitivity to blue wavelength helps here; reference needed)
- it helps to separate cold and warm regions of the image
From the computational point of view they are independent.
They affect each other, but it does not matter which one you move first, the result is always combined from those two settings. While the temperature is used in a creative way and may be almost artistic, the tint slider is mainly used to correct lighting imperfections in the image and is more of an "emergency image saver" than a creative tool (even though in some cases it might be used creatively too). In most cases, I let the CameraRaw set the tint automatically (by selecting a white point in the image, using the tool that has color picker icon with target beside it) - that makes sure I corrected unwanted color changes caused by colored walls, neon lights nearby etc. When this tint setting is done, I then play with the temperature as I wish to get desired atmosphere.
The settings doesn't actually affect each other directly, but if one is too much off in the wrong direction, it's hard to see what the other should be.
Personally I find it much harder to adjust the tint, so for tricky images I first set the temperature to a setting where it's easiest to see tint changes, adjust the tint, and then set the temperature.
For example, to adjust the tint right for skin tones (caucasian), I set the temperature way too warm so that the people look mostly like Simpsons. That way it's easier to balance the red and green.