First, an important point for any ensemble playing...
It's YOUR job to count the rests.
Or, put another way...
It's your JOB to count the rests.
Now, a tip, and hopefully a discussion starter. Some people seem to be able to keep track of long stretches of rests in their heads without trouble, but I'm not one of them. I use my fingers. I use a particular method; I don't think it's terribly original, but I'll share it here to see what others think.
I count to 10 on one hand, starting like this:
My thumb goes out last for 5, then I start with the thumb for 6, putting my pinky out last for 10:
Then I start again with the pointer for 11, etc. So, all I have to remember is if I'm in the 10's, 20's, etc. (The longest stretch in Così was 69 bars!)
I do this discreetly, in my lap where the audience won't notice. And sometimes I'm not even sticking my fingers out as in the pictures above, but rather I'm just pressing one finger at a time into my thigh or my horn to keep track of the count.
One last point related to this topic: If you know the score really well, you don't even have to count every rest.
I would like to know what methods you use to count rests. Please describe them in the comments section!