How many transactions need to include fees?
None, technically. If there's even just 1 miner out there with a reasonable hash rate at that time that feels like including a transaction with zero fee in a block, then it'll get included. That even happens right now, it's just that they get prioritized away in favor of those with fees or those transactions that transact a larger sum.
How much should those fees be?
So let's assume that one should in fact include a fee, and for sake of argument, imagine that only a single transaction is included in each block - then the fee for that single transaction should equal or exceed the expense of mining the block. But then here comes your third question:
Would major players in the mining industry pack up and leave after the last block gets mined?
Probably. Maybe not, but probably. But when that happens, the overall network hash rate drops. Which means the difficulty drops. Which means that it becomes relatively more profitable again to keep a miner running. Ideally, some state of equilibrium would naturally get reached. Less ideally, the more efficient hardware or hardware hosted in areas where e.g. cooling is free and/or electricity is free will have an advantage (as is already the case now, just not as pronounced as it would be then).
Mind you, 'the last block' is a very long time from now - you and I will almost certainly be dead. I'd say wait and see what happens come mid-2016 when the block reward is halved.. it will be an almost unique time in mining in that it will be an abrupt change that is relatively large (the one from 50 to 25 was larger, but mining was in its infancy anyway - while the one after is 'just' from 12.5 to 6.25) while mining is still likely to be a very busy industry (even if it all moves to giant farms) so we can see how the mining market reacts to that one.
But even that is 2 years out - and 2 years is practically an eternity for mining. If you remember the Block Erupter USB - that was introduced a little over 1 year ago. Who knows what will happen in that time alone - never mind by the time 'the last block' is mined
So the height of the fee hinges on what the operating expenses would be for miners that would mine that block, and if you can predict that - you should probably be in the business of predicting world cup match outcomes instead
