When completing challenging workouts on Trainerroad I am often confronted with a situation where I completed an interval that felt near impossible but have many more to do. My question is, if I'm not quite feeling up for the next interval, is it better to give myself a few minute longer break in between? Or simply lower the intensity for the subsequent intervals?

Honestly it depends on the specific workout and the type of intervals. Some intervals, such as shorter on/off VO2Max intervals you're not meant to fully recover and so taking extra rest would be 'cheating'. Better in that case to drop the intensity a little. On the other hand 3-5min max intervals a little extra rest might be OK.

I've also found that some TR workouts, the rest interval may be longer or shorter than ideal simply because they'retrying to fit a specific amount of work into a specific duration workout.

You make some excellent points and I see what you mean concerning efforts like VO2 max, but if a VO2 max effort is say, 110 percent FTP and I'm feeling terrible and decide to drop it by 10% I'm at 99% FTP which basically turns the interval into FTP training. Would this still be preferable for these types of intervals instead of a break? Or would you be more likely to only drop the intensity by a figure like 7-8% and add a small (say 30 second) break to ensure it stays in the "VO2 max" range?

Well you don't want to drop the intensity so much that you change the type of interval. In your example of 110% VO2 intervals, I would try dropping to something like 105-107%. If you still can't do that, it's time to bail. Either you're too tired for the workout, or your FTP is not set correctly.

If you're not making your targets, you aren't going to produce the adaptations you're after, and you're just digging yourself into a recovery hole. Mind you there's a big difference between "balls to the wall" (barely structured training, no real goal) and a specific prescribed power output.

Like others have mentioned, it depends on the specific intervals. Zones are not steps so adaptions occur at different rates in various zones so you can still get some of the benefits at a slightly decreased intensity, but for V02max+ type of work you need to be in those zones to get those specific adaptions