That's extremely unlikely for two reasons. First of all, the smell and lack of hygiene. Imagine several hundred or thousand people urinating in a theater during a performance. On a hot day in summer, which Athens has a lot. That really stinks to the sky. Just that alone would be a good reason for not doing it.
But urine was a very valuable commodity. It was used in many chemical processes. For example, tanners needed urine - lots of it. It was used by launderers, for whiting textiles. And much more.
Urine would be collected. Shops and taverns had bowls where you could relieve yourself in. The contents would be sold to the above industries. That was such a profitable business that Vespasian taxed it.
I don't think there would have been any real difference between the way Greeks or Romans behaved here. Urine was just as valuable for Greeks as it was for Romans. If a Greek theater didn't have a place to relieve yourself, entrepreneurs would make arrangements for that next door.