I am posting here as a Cinema Manager and so I would like to express my opinion and the happenings in my own place of work.
People who quoted info about the ticket prices are correct. Cinemas, although in the past they made their money on ticket prices, now need to use their shops and concessions stands to make their profits. Every film that enters a cinema has very low returns for the company nowadays, as the large movie companies want a larger share of the cash.
And so cinemas buy in foodstuffs at low prices, and have a massive mark up on said items. This is why they use the Policy of 'Only food purchased in the premises may be consumed in the auditoriums.'
If you look at the various sites, you will see this stated in their policy - often a piece of laminated A4 paper stating other things such as the management have the right to refuse, etc etc.
Now, in my own cinema, we use better judgement. Personally we side with people that food is way too expensive if you have a large brood. We have no problem with people bringing in their own snacks and drinks, except large bottles of pop which we ask them to leave behind the counter. We give them cups and tell them to come in and out to refill, as many used to bring in large bottles just to spill them and have us cleaning up.
We allow bags of all sizes, however I know Dun Laoghaire have a policy of no plastic carrier bags, for child safety reasons.
We also won't allow hot fast food into the screens, as they really stink the place out. If a large group of teens come in, we also ask that they leave their bags in the store room, to ensure that people aren't tripping over them, etc etc.
So, the original question. Management do have a right, from site to site, when these conditions are displayed to ask that you leave food behind the shop. Theres no law as such, but when you display conditions, people entering the premises are expected to abide by these rules.
However, if you popped along and spoke to a member of staff, you may find that you have no problem.