I show jump and I have my team out every second weekend (twice a month) I normally do 3-4 day shows (not every horse is ridden every day) then they just get one day off and lunged the next. I do that for about 3-4 months then give them a week off around Christmas when out show season is then they get another 3 months off at the end of the season :) it doesn't matter how little or how much time you give your horse/s off as long as they do get a break once a week or after a show :)
For my barrel horse, he gets 1-3 days off after a big race
I would ride in shows two months and then I would give my pony a weeks break
For my horses who do a full show season, I try to really separate the show periods : I would show for 2 or 3 weeks in a row, than let them rest without showing for 3 to 4 weeks. They also have a real break in winter of 2 months without jumping a single fence at all (December/January). Every week, they have a day completely off when they just go to the treadmill or paddock. We try to flat work in a constant way, to avoid long sessions to catch up the work. They also go outside a lot to see something else than the sand arenas! So I think that there are really 2 things to consider when you have to keep your horse in a good health: physical and mental tiredness :)
If your availability does not allow you to work every two days, it is necessary to juggle to balance the week: a "big" session can be followed by a session of "stretching" for example.
Or then a rest day or of paddock :)
It is good for his morale to vary his activity (work on foot, course, rest, meadow etc.) So I think that one or two rest day a week is good if you don't do big sessions every time ;)